You can battle by finding gyms (big colored towers with a pokemon on top of them) after your trainer hits level 5. The strength of the gym varies based on who fought it previously and the buildup of the teams in your area. The range in my area for example typically goes from about 300 cp to 1200 cp per pokemon, and we're mostly dominated by blue colored gyms.
Speaking of, your dynamics with gyms depends on your team color. You get to choose your team after you attempt to challenge a gym after level 5. If you are fighting a gym of a different color, you may bring up to 6 pokemon regardless of how many pokemon are in the gym. If you are fighting a gym of the same color, you may only bring one pokemon with you at a time, once again regardless of the opposition.
You may evolve pokemon by meeting their requirements. When you capture a pokemon, you are given the option to power it up (which just raises their cp, the combat power of a pokemon) or evolve it (which, of course, evolves the pokemon, and also grants +100(?) cp). Powering up is easy, as you typically have to pay a cheap combination of stardust and pokemon candy, which you get both of just for capturing the pokemon. Evolving a pokemon is far more difficult, as it requires a vast amount of pokemon candy. To put things into perspective, you get 3 candies of the base evolution of a pokemon whenever you capture a pokemon. You can transfer the pokemon to your pc (in game pc, not irl pc) to permanently release it from your poke bank and get 1 candy. Pidgey for example requires 12 Pidgey candies to evolve. Pidgeotto however needs 50 Pidgey candies.
I am quiiiiiiiiiite sure this is not the whole game. For example, we only have Generation 1 Pokemon, and there is a rumor about directly battling other trainers in a later installment of the game. Give the game time to secure itself and grow and we'll probably see some more interesting developments down the line.