Look for the blue spots on your map. They give you Pokéballs and eggs and such, letting you catch everything you come across even if you don’t really need it.
Once you hit level 5, you can start going to gyms. You can pick from one of three teams. Depending on which team you join, you can reinforce certain gyms and help other members of your team. As of now, the gym leaders are generally pretty weak, so don’t bother leaving your strongest Pokémon at a gym.