In Tony Hawk's Pro Skater (the first one, for the PlayStation), is there a way I can check the special moves for the skaters in-game, or do I have to guess them myself? I know I can just search online, but I want to know whether there's a way to do it in-game.
Not in-game no. The very first game was pretty barebones compared to what it is today. There's no move listing, gap listing, trick editing... nothing. You had to find out what the tricks were yourself (per skater) or refer to a guide.