Best free things to do in Saint Petersburg (Florida)?

Most people who come here spend a lot of time at the beach, and if that's what you're looking for as far as outdoors, Pass A Grille Beach is good for a chill time, PCI (Post Card Inn) is more of a wild scene with drinks. 
Now that the beach is out of the way, the real Saint Pete Beach is not to be missed. That includes the downtown area on Beach Drive Northeast (upscale, pricey restaurants, but a beautiful area to walk around with a nice park full of huge Banyan trees by the water), Vinoy Park, St. Petersburg Fl (almost always see dolphins here because they use the wall to fish), and the block around Central Ave and 1st St. N, which is always active, every night of the week, and always has cheap drinks. Further up Central Ave, before 34th St. is the Arts District with lots of breweries, coffee shops, etc. The The Dali Museum is our most famous but can be pricey unless you're a student and the same goes for most of the others: the Florida Holocaust Museum, Museum of Fine Arts, Chihuly Collection, and the Saint Petersburg Museum of History.
The best park to see here in St. Pete is probably Weedon Island Preserve. There isn't a lot to do outdoors here unless it involves water, which typically involves equipment rental. My best advice is to check Groupon for kayaking, SUP, etc. Sunken Gardens almost always has a $5 Groupon and worth that much, at least.
If you have time to head north a bit, Florida actually has some of the most pristine natural springs you'll ever see. Tubing on Rainbow River Area is about 2 hours away and might cost $5 to rent a tube. Kayaking it is worth the $40 or so as well. At Crystal River, you can swim with manatees in their natural habitat. Again, check Groupon.
Enjoy!

I would visit the new pier. There is a shuttle or you can enjoy the walk. Street musicians, grass area to bring a picnic, and great people watching.