Benefits of training:
Battling alongside your team is one of the many things you can enjoy here. Battling other pokemons helps you to gain experience and climb up the trainer rankings. Training increases your trainer ranking and helps to build up the levels of your precious pokemons. Not only do they give you trainer experience, they are also the most efficient way of gaining battle points and cash. Battle points can be used in the BP shop to buy cool looking pokemons ,whereas cash has high significance for buying stuffs off the mart co. and department stores in game.
There are many options of training , starting from player vs player to npc fights to team rocket.
The information on these can be obtained here: NPCs
All these trainers can give you a good amount of experience. The higher the opponent's pokemon level, the more experience you will get per battle.
Most efficient way of training:
The best place to gain some easy and quick experience is to fight the dark trainer.
You can battle the dark trainer once you have obtained all the gym badges. The dark trainer has 6 pokemons:mew, mewtwo, shinymew, glittermew, frostmew and goldenmew. All these pokemons are 9x the level of your highest pokemon. It is pretty easy to defeat the first two pokemon. You need to buy recover and toxic from the store and make your pokemon learn the moves. The best choice of pokemons to use against the dark trainer are ghost type pokemons like golurk, giratina, etc. Ghost types are almost vulnerable to the moves used by mew and mewtwo. So you can poison them using toxic and keep using recover until those pokemons faint. Once you have defeated the mewtwo , you can restart the battle and keep doing so if you are fond of long hours of training. If you want to defeat the dark trainer's all pokemons, well it is not as easy as you think. This is because the shinymew and glittermew have moves like seed flare, surf and thunderbolt which can send your ghost types back to your pokeballs. But if you manage to defeat them by saving your main pokemon( using toxic and switching to other pokemon) , you can easily defeat goldenmew and frostmew as they have movesets almost similar to mew and mewtwo. If you defeat the darktrainer, you will receive a fossil. Battling against dark trainers will help you rank faster than other methods.
You can also capitalise on the 2x exp events on sundays and other exp events on special events to gain more experience .Equiping the pokemon you want to train with lucky egg will give 1.35x more experience.They can be bought in mart co. or can be found equipped with wild chanseys in maps. You can also use exp boosts to gain more experience per battle.
So, that's it. Good luck training and enjoy the game.