If you are talking about a small helium balloon (like the floating kind used at birthday parties), you would use “launch” or “Let’s let the balloon go”. or “let’s let it fly.” You wouldn’t say “fly the balloon” because “fly” suggests that you are in control of the balloon somehow, which you are not. If you are talking about the kind of huge hot air balloon that can carry people, “Let’s launch the balloon” could be used to refer to the process of lifting off from the ground, while “Let’s fly the balloon” could refer to the whole process of traveling in the balloon (that is, both the initial lauching and the later floating and landing).

They have slightly different meanings. "Launch the balloon" means getting it off the ground. "Fly the balloon" means having it in the air for a while.

'Launch' sounds too serious (like when 'space agencies' say they're sending a rocket to space.. then it flies over the ocean, out of sight, ha ha). so, you should say 'let the balloon go'. You can say 'fly a kite' but I don't think it's normal say 'fly a balloon'... I don't know what I would say, though.. :)