The City of Dallas Office of Arts and Culture runs four cultural centers that have lots of free programming. Most of them have an art exhibition of some kind going on at all times, as well as other things on their schedule. They each also have interesting things nearby.

The Bath House Cultural Center is near White Rock Lake (which in and of itself is free and fun to hang out at.)

The Latino Cultural Center is near Deep Ellum, a pretty short walk from the Baylor DART rail station. A few blocks away there's a neighborhood of charitable organizations with offices in old houses along Swiss Avenue that's an interesting walk. You'll come across Central Square Park, which has a little Gazebo in it - a charming spot for a picnic. You can also walk a few more blocks to Exall Park, which is a much larger park that has a couple of cool scultpures in it (as well as regular park things.

The South Dallas Cultural Center is near Fair Park. Wait until mid-November (after the fair is over and has moved out) and then you'll have all of Fair Park to walk through. Visit the African American Museum.

The Oak Cliff Cultural Center is in North Oak Cliff, several blocks south of Bishop Arts. The Oak Cliff North Branch Library is just a few blocks away. Continue a little further to Bishop Arts and hand out at The Wild Detectives bookstore.

Bookstores are also great places to hang out for free. Join or drop in on book clubs, go to book signings/discussions (you don't have to buy the book), and check their website or socials for other events. Just wandering around is fine too. Half Price Books is probably the most entertaining to just walk around and explore, but Deep Vellum, The Wild Detectives, Whose Books, and Interabang Books have numerous events on their calendars.

The Dallas Farmers Market is fun to walk around, especially right now while the Pumpkin Patch is going on. (They do nice decorations for Christmas too.) They have occasional free entertainment/events on their calendar as well.

If you enjoy public art, there are works at most of the 30 Dallas branch libraries and at all of the 65 DART Rail stations. Here's a guided public art walk map for downtown. If you want to explore murals, there are tons in Deep Ellum and Bishop Arts, and a fair amount in Uptown and the Design District, and don't forget the Fabrication Yard. Pick where you're going to go, and do a Google search to find a list/map/guided tour. (Lots of blogs have these.)

Downtown parks have free activities - there's pickleball night, yoga classes, movies in the park and other things - at Harwood Park, Carpenter Park, Main Street Garden Park, Pacific Plaza, and West End Square. I use the Downtown Dallas Inc calendar to find these events. (Uptown Dallas Inc has a good calendar too for things like movies in Griggs Park.) Thanks-Giving Square is also pretty cool. I think everyone should visit the chapel at least once to see the amazing stained glass mural inside.

Dallas Parks and Recreation calendar has occasional events on it. There are Volunteering/Beautification events at various parks and community celebrations at various rec centers. It's packed with Halloween and Dia de los Muertos events right now.

Along with the recs others have made in this thread, these should keep you busy for a while, but also remember, most of these are not "one-and-done." You can go to the main branch library and Dallas Museum of Art every month and see something new!