- Refer to it as St. Looey for fun - lots of people call it this - not all.
St. Louis is secretly awesome - the downtown is an under-utilized skyscraper forest, but outside of that the people are really cool in a unique way (the good kind of hipster) and will appreciate hearing about your adventure etc. Restaurants will cheerfully give you free food if you plead hunger and offer to scrub dishes for your supper- they actually won't want you to work, but they will probably give you some kind of free food in takeout form- actually you should give this eating scheme a shot any time you're hungry in the US- it's just especially good in St. Louis.
By all means go to the City Museum! Words cannot explain! It's not really a museum, it's more of a 10-story ex-shoe-factory-turned-post-industrial architectural relic sculpture that one can walk, slide, and climb all over. Its made out of discarded materials from the remains of the dearly departed industrial era of St. Louis City ("First in booze, first in shoes and last in the American League"). Its all welded and cemented together in very imaginative ways. A highly eccentric and amazing artist named Bob Cassilly and his hand-picked team of artists made it. It has a skate park, a thrift store, a cavern with turtle sculptures, a model train setup- post-apocalyptic postindustrial wonder. If you have no money just plead poverty to the people at the entrance where you're supposed to pay and say we hitchhiked all the way here from (wherever) just to see the city museum and ask them for a temporary pass- with the temporary pass it's free to see the museum, they'll just try to kick you out after 15 minutes. Simply remain as long as you can.
I feel like if you paid (so they weren't looking for you after 15 minutes) you could discretely find a place to evade the closing-up patrol and sleep the night there- that would be awesome to be in the city museum all by yourself- I hope some people give this a shot- we got kicked out after about 30 minutes (we got temporary passes) and departed by way of a 3 story slide (there's another slide that goes all 10-stories from the roof to the ground floor that workers used to throw finished shoes down). Happy adventuring!
Also free and worth a mention in the St. Louis Downtown is [the Wainwright Building], one of the world's first skyscrapers, The Gateway Mall, a multi-block fountain and sculpture park in front of the Arch, and, if you want the tourist pic with the city skyline behind you, there is the platform across the river from The Arch at Malcolm Martin Park. There is a walk-up platform, there.