Get away with 5/4“ cedar decking across a 24” span in a children’s play structure

The Scout Pioneering website contains information on how to build several bridges. I am guessing that most of these are more complicated than what you had in mind, so if you want something simpler, you can attempt flat span bridge using 2x12's for a span between 14 and 18 feet. On the latter page the author goes on to say that "[i]f you are thinking of building any bridge with a span greater than 20 feet, you better get a structural engineer involved, plain and simple", which I am inclined to agree with.

Note that none of those links say anything about the actual load-bearing capacity of the bridges, but I would assume that most of them would work fine for one or a few people crossing at a time.

If you want to calculate maximum load, it is complicated, especially as the materials in the bridge are likely to deterioriate over time - especially if the environment is wet. Some simple calculators for point loads and distributed loads can be found online to give you an idea of what your bridge can take, but as they make clear on their website, they provide no guarantees. In other words, if you want someone to guarantee you that your bridge is safe, you'll most likely have to hire an engineer, as circumstances change from bridge to bridge.

A swing frame must be very rigid or the energy put into the swing gets sapped, causing it to quickly lose momentum. This is why you rarely see swings on posts. I'd stick with a proven design, the A frame. If you really must use vertical posts, go at least 3 feet down. With 10 feet above ground the torque will be quite strong.

Interior diagonal braces effectively reduce your beam span. You could easily cut 3 feet from each side, leaving 10. I'd trust a select 6x6 to span that just fine. The braces must be securely fastened, though. I'd consider mortising them into the beam and posts.