Pokémon GO: 15 Crucial Tips & Tricks The Game Doesn't Tell You
3. How To Battle Other Players & Becoming A Gym Leader
Okay, you can't actually fight other trainers in real time - but the nearest implementation comes from Gyms.
First you'll need to level up to 5, and then claim a neutral gray-coloured one for yourself, or head up to a coloured gym (meaning it's owned by members of the three teams, Instinct, Mystic, Valor), as therein you'll get the option to battle their Pokémon one on one.
Depending on how many real-world people have staked their claim at that specific spot, the person with the highest combined CP (Combat Power) will be the default 'Leader' (who you'll face last), with all others simply being trainers you need to get through on your way to the top.
Note, battles are NOT turn-based. Instead, you swipe left and right to dodge and can tap on your opponent to do as much damage as fast as possible. The only 'tactic' outside of evasion is keeping an eye on your special meter, but chances are the fight will be over before it's maxed out. After you win, depending on how many people you need to get through, the gym may still be coloured - so repeat the process until it's gray, and then plant your metaphorical flag.
Once you do, you're then free to be the Leader of that gym, providing you can keep your CP high enough.
It's rumoured on various forums (with no sources yet) that Niantic are adding turn-based, real-time battles in the future, but for now, head out, amass your army and dominate as many gyms as you can find!