What can I do with trainer levels?
There are different rewards as you level up:
As your level increases, so will the CP of wild Pokémon (which means you can capture stronger Pokémon) and you can find more species.
Your level determines the CP cap for your Pokémon, so higher player levels mean that you can power up your Pokémon to higher CP.
At level 5 you can start challenging gyms and participate in battles.
At level 20, you receive Nanab Berries instead of Razz Berries. Then every odd level after 20 it will alternate between Nanab and Pinap Berries as a reward. source
Each level up grants item rewards, here is the list:
Level 1: (Starting Level)
Level 2: Poké Ball x10
Level 3: Poké Ball x15
Level 4: Poké Ball x15
Level 5: Poké Ball x20; Potion x10; Revive x10; Incense x1
Level 6: Poké Ball x15; Potion x10; Revive x5; Egg Incubator x1
Level 7: Poké Ball x15; Potion x10; Revive x5; Incense x1
Level 8: Poké Ball x15; Potion x10; Revive x5; Razz Berry x10; Lure Module x1
Level 9: Poké Ball x15; Potion x10; Revive x5; Razz Berry x3; Lucky Egg x1
Level 10: Poké Ball x20; Super Potion x20; Revive x10; Razz Berry x10; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1
Level 11: Poké Ball x15; Super Potion x10; Revive x3; Razz Berry x3
Level 12: Great Ball x20; Super Potion x10; Revive x3; Razz Berry x 3
Level 13: Great Ball x10; Super Potion x10; Revive x3; Razz Berry x3
Level 14: Great Ball x10; Super Potion x10; Revive x3; Razz Berry x3
Level 15: Great Ball x15; Hyper Potion x20; Revive x10; Razz Berry x10; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1
Level 16: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5
Level 17: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5
Level 18: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5
Level 19: Great Ball x15; Hyper Potion x10; Revive x5; Razz Berry x5
Level 20: Ultra Ball x20; Hyper Potion x20; Revive x20; Nanab Berry x20; Incense x2; Lucky Egg x2; Egg Incubator x2; Lure Module x2
Level 21: Ultra Ball x10; Hyper Potion x10; Revive x10; Pinap Berry x10
Level 22: Ultra Ball x10; Hyper Potion x10; Revive x10; Razz Berry x10
Level 23: Ultra Ball x10; Hyper Potion x10; Revive x10; Nanab Berry x10
Level 24: Ultra Ball x15; Hyper Potion x10; Revive x10; Razz Berry x10
Level 25: Ultra Ball x25; Max Potion x20; Revive x15; Pinap Berry x15; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1
Level 26: Ultra Ball x10; Max Potion x15; Revive x10; Razz Berry x15
Level 27: Ultra Ball x10; Max Potion x15; Revive x10; Nanab Berry x15
Level 28: Ultra Ball x10; Max Potion x15; Revive x10; Razz Berry x15
Level 29: Ultra Ball x10; Max Potion x15; Revive x10; Pinap Berry x15
Level 30: Ultra Ball x30; Max Potion x20; Max Revive x20; Razz Berry x20; Incense x3; Lucky Egg x3; Egg Incubator x3; Lure Module x3
Level 31: Ultra Ball x10; Max Potion x15; Max Revive x10; Nanab Berry x15
Level 32: Ultra Ball x10; Max Potion x15; Max Revive x10; Razz Berry x15
Level 33: Ultra Ball x10; Max Potion x15; Max Revive x10; Pinap Berry x15
Level 34: Ultra Ball x10; Max Potion x15; Max Revive x10; Razz Berry x15
Level 35: Ultra Ball x30; Max Potion x20; Max Revive x20; Nanab Berry x20; Incense x2; Lucky Egg x1; Egg Incubator x1; Lure Module x1
Level 36: Ultra Ball x20; Max Potion x20; Max Revive x10; Razz Berry x20
Level 37: Ultra Ball x20; Max Potion x20; Max Revive x10; Pinap Berry x20
Level 38: Ultra Ball x20; Max Potion x20; Max Revive x10; Razz Berry x20
Level 39: Ultra Ball x20; Max Potion x20; Max Revive x10; Nanab Berry x20
Level 40: Ultra Ball x40; Max Potion x40; Max Revive x40; Razz Berry x40; Incense x4; Lucky Egg x4; Egg Incubator x4; Lure Module x4
Level 40 is the maximum level
My source seems to be updated often (yesterday it only had reward info up to level 12)
Edit: Another source
Here's a nice table that summarizes the rewards through level 20:
It appears that the level cap is currently 40, with the following rewards:
Ultra Ball x40
Max Potion x40
Max Revive x40
Razz Berry x40
Incense x4
Lucky Egg x4
Egg Incubator x4
Lure Module
You can find further info and relative rewards on this Reddit or on Serebii.
Level 5: you can join teams and get to do gym battles.
Level 6: you can get a new type of Incubator with 3 uses, allowing us to incubate and hatch several eggs at the same time.
Level 8: you can get razzberries, which make the next ball you throw have an increased chance of cathing the pokemon.
I've not found any other information regarding leveling up, but I will update this as new information is presented.