Right. There are eight gyms in each region. You're likely in Johto right now, unless you took the ferry. The goal of a Trainer is to defeat each gym leader's team, and they all specialize in certain types. The first one is a flying-type gym, so I'd advise you to look for Pokemon with electric, rock, or ice moves. Defeating them gets you money and a few other items, as well as the ability to use special items known as HMs that teach your Pokemon moves that are commonly used outside of battle.

Why you can't fly on a bird or surf on a giant turtle without a badge that says you can, I don't know.