What are some free interests/things/hobbies you can do in the city?

My first thought is going for walks around where you live. If there are green spaces along the way that's even better. You can turn it into kind of a workout where you do longer and longer walks, or the same length but a little faster each time.
It's a good way to clear your mind, it also never hurts to be active.
- Some more variants if a workout isn't your thing
  take photos on your phone when you see something nice
  contribute to OpenStreetMap by doing quests on StreetComplete (you answer questions about things you walk past)
  Geocaching, if you live somewhere with things to find
  
  I didn't know the StreetComplete app. It's so cool ! Thanks !
  
  Pokémon Go is also an option!
- Mix it up by walking to local landmarks and public art.

Look for volunteer opportunities. In my town I found a litter-picking group that met once a week. Then through members of that group I joined another one that maintains flower beds and planters around the town. Then joined an effort to rehabilitate an environment project on a nearby farm, and ended up in the beekeeping team. Another group I was in for a few years organises gentle walks for elderly folks. I learned a huge amount from all these things, and none of it cost me anything but time.

The city where I live has a musical instrument lending library. I don't know how common these are? Ours started when a cherished local musician passed away and his eclectic collection became the library. Over the years, more people have donated instruments and there is an annual festival to raise funds for their upkeep. (As a local musician, I'm actually playing at said festival today.)
Anyway, it works just like a regular library. You get your library card and check out an instrument and it doesn't cost you a penny. And there are all kinds of videos online these days to give you pointers on how to play. I guess if you get really serious, you'll probably want some one-on-one tutoring, but if you're just doing it for kicks and don't have any plans to join a band or whatever, you can just have some fun and see how far you can get on your own?

Groups and clubs. They exist, you need to find them. The public library is a good place to start looking.

I like going for a walk around the city or to the Rhine river and watch the ships go by.
If you’re the social kind, go to the park and ask someone who plays frisbee or other activity to join in.
Running is a great sport that’s relatively cheap and also helps against depression.
Geocaching is fun alone and with other people, but not in the city (in my opinion).
Check your local newspaper and see what events are going on. Sometimes events are cheap or even free.

Have you got a seat filler company? It's a company that you pay a small amount to go to gigs, theatre, events, etc that have been undersold. They just want people to fill the seats. I pay around £3 a time and I've seen bands like Kiss, Less Than Jake, Fun Lovin Criminals, not free but cheaper than a coffee.
- How do you find one?
  
  Google? Seat filler + name of the city

chess or other board games are inexpensive and you can learn a lot and make friends along the way.
Fotography can be expensive, but if you do it just as a hobby, you could take your phone or get a cheap camera and try to get up early on weekends and take some photos of the city.
reading books, especially about animals (specifically birds) could be an option too.
- I agree on board games.
  See if there are Board Game shops in your city. If they have tables for Magic the Gathering you can check their calendar and usually find open board gaming nights. In my experience people usually bring a bunch of their own games and are open to new players. Some of the people in our group don't own any games; the rest of us are always bringing more than we can possibly play anyway.
- So just like take a chess board to a park or something?
  
  If you see other players. Probably less awkward to try to find a club.

Search for
Boardgame club (insert your city name here)
Usually you'll have a handful to choose from.
The ones in our city are:
$5 entry
Last for 10hrs
Go with friends or go alone and meet people on the day who want to play a boardgame
Mostly friendly/helpful people who want to teach and enjoy games with others
Choose your level of social interaction (no one will bat an eye if you only talk to play the game, and make no side chit-chat)
Fun everytime

Geocaching is free and usually lots of fun in cities. It's like a big database of dead drops - people hide small containers with pieces of paper to sign, and post their GPS coordinates online. Frequently they're hidden near points if interest, as well so you might find some cool shops or bars as a side effect.

Frisbee golf, biking, rollerblading, skateboarding, gardening, cooking, painting, geocaching, hiking, fishing are all pretty cheap to get into.
- I want to second cycling. It's a good way to explore your city for free as well as getting shape. There are often cycling groups that you can join as well if you want to socialize on top of it

CrossFit, running club, November project, hiking club, board game clubs, DND clubs, Meetup.com events. Coed sports leagues like: disc golf, infinite Frisbee, soccer.
There's also things like live figure drawing, music jam clubs, acting in local plays.

Community theater is great for meeting people and usually basically free
If you have a university in your town often they will have some clubs that are open to the whole community and occasionally public lectures.
Museums and art galleries are usually pretty cheap especially if you keep your eyes open for the free stuff. I go whenever I see free stuff, or a new exhibit I'm particularly interested in.
I know people that do the indie concert scene, just going for any random concert under 20 bucks. Sometimes the bad ones can be just as entertaining as the good. you go to maybe 1 show a week and spend the rest finding the concerts and maybe listening to them on spotify/sound cloud.
Indie movie theaters can also be a vibe. Mine is about 10bucks a ticket, but sometimes they have a free night and sometimes a pay what you can deal. I go every month or so. Last one was a scooby doo one, it had a "cheer whenever they say scooby" game, a drinking game and the previews were funny clips about scooby (robot chicken and that sort of thing) it is a very social experience and a lot of fun. Not free but clearly not for profit.
I think the best thing is to just walk around more, and just be on the eye out for stuff that interests you. Actually read the fliers on posts, join local social media groups to find out what's up, keep an open mind and look to push your comfort zone and look out for local community stuff over corporate for profit stuff.

We have a simply gorgeous greenhouse that is free to the public every day in the middle of a wonderful park.
Our city library cards also give you free access to all the city art galleries, which is a wonderful thing to do. You should check out your city library to see what they offer.

Top tips: consider cooperative living and take group classes for one of your hobbies

Try to identify your local ethnic/cultural organizations, and follow their events schedules.
There are pretty large Indian and Vietnamese populations where I live, and they each have an association to share their cultural highlights with the larger pop. The Holi and Festival of Lights celebrations (Indian) and Lunar New Year (Vietnamese) were all an absolute blast and had amazing food.
Good way to get intel on your local organizations is to hit up the respective restaurants/grocery stores from whichever cultures are well represented in your area - look for pamphlets and ask the staff.
As a pasty white dude who's only recently started tuning in to events like that, everything feels very new and unique - highly recommend!
- This! My area has a sizable Hispanic population and I wasn't aware of how many street festivals they have until I moved here. It's pretty great. Had the best churro ever the other day, and got an amazing poncho for day off the dead.

Photography is free if you have a smartphone.

Id recommend disc golfing. You can start if with just a mid range disc and that'll cost ~$15 then your good to go. Most major cities will have at least one course that's free at a park so you don't need to spend more after that. It has a pretty low skill floor so you can pick it up pretty easily but a very high ceiling so there is a lot to learn and grow which can help with depression. Also gets you walking outside in nature which can help with depression too. It can be as social as you want it to be, you can invite friends, or just go solo, and even if your solo you can strike up conversation with the people in front of you and sometimes they'll even let you throw with them.

Pick up sports, or adult leagues. It can be hit or miss, but I made some good friends playing in rec soccer.

Find a game store! Every city should have one or two places where people gather to play board games and card games. I've met some really great people just jumping into random magic the gathering matches. Only of my best friends was just some dude I met at the card shop, he was looking for a group to play dnd with and we've been meeting up once a week ever since.
Apps like meet up (if it's still a thing?) may be really useful as well.

Maybe getting a gym membership, if it's cheap and you're interested?
I've not gotten one, but have heard of friends n others who found decent friends in the gym.
And the exercise may help with the general mood too.

I suggest you read "The Game: Penetrating The Secret Society of Pickup Artists” by Neil Strauss. Not to become a 'pickup artist', which itself is a joke and a very shallow life even if one were "good at it". No.
However, there is a chapter or two, I can't remember how much but it could have been half the book, about self-actualization that I thought was very useful. About making your life full, so that you have a rich background of things you enjoy doing to bring people to.
It's been forever since I read it, but it's an interesting read, especially when Courtney Love is being den-mother to a houseful of pick-up artists in training. Neil Strauss can tell a story. He also wrote a biography about living with Mötley Crüe that is pretty depraved.

skateboarding
a decent starter one will run you $100 at the most, pads can be found for cheap; all you need is a flat surface and a non-rainy day. you will fall a lot but enter into a worldwide community that has its own very real culture. keeps you fit, gets you outside, lets you meet people. best investment in my life

The city where I live in has several few Little libraries that people have in their garden or hanging at the wall of their home. And I know that other cities have them as well. Today I went biking and brought back two books and I was pretty sure that I did not want to take any because I have more than enough to read, but I noticed an interesting book and took it home. This is something you can do yourself. Build a small wooden box, and have other people put or take books. You don't have to do this yourself, you could ask friends, family and co-workers to help you build it or ask them to provide such. Then you could take a little bit care of it and promote the idea.
What I like about bigger cities are public parks. I like to go to parks and just sit, listen to birds, enjoy the green and in Springtime, Summer and Autumn enjoy the colors.
Public libraries here have magazines and books to read for any visitors. I've learned about open source software thanks to the Internet but also a little bit thanks to public libraries. I find that reading paper books or magazine can also be a nice break from staring at a screen of a device.

Pokemon Go. It will entice you to take long walks in areas that you may otherwise not visit. I moved from a rural area to a big city and play Pokémon Go daily with my pup, rain or shine. We end up spending hours outdoors and it's done great things for my mental health.
- nice

Library.

Drawing / urban sketching.

Sabotage an electrical substation! Punch a fascist!
(/s I don't condone violence)

I like to go to art craft market sales. Not the school gym type, the weird stuff on the sidewalk type. There's usually interesting stuff and people to see, and sometimes food.

Depends on the country / city but (some) art galleries / museums might be free or have free days ...

Might help if you name what city you are in. If there is a college there are often cheap/free things you can latch onto even if you aren't a student.