Catching pokemon gives you XP and Stardust and sometimes candies.
When you catch pokemon, if you scroll down to the bottom of the catch screen, you can transfer them to get additional candies.
When you select a pokemon, you have the option to power them up, the first button, which uses stardust and candy. When you power them up, they gain more CP points making them more useful in battles.
The second option is evolve, when you gain enough candies you can evolve a pokemon. (Tip: if you catch lots of weaker evolution pokemon, like weedle's and pidgey, they only cost 12 candies to evolve and give you lots of experience for each evolution. This can help you level up your account faster so that you gain better bonuses and items.)
And of course the last option is the transfer button at the bottom.
To explain the pokemon a little bit better. As you get up higher levels you will get stronger pokemon. If you evolve a 81 CP pokemon, the resulting evolution will be between 130-210 CP(this isn't an exact science) However if you evolve a 250-500 CP it could go up to 500-1000+ CP as you level up, the evolutions have more impact also.
With the pokemon sizes, XL generally has higher health I've found, and also takes less damage(I'm guessing higher Defense), it doesn't of more damage as far as I've noticed though. XL pokemon dodge slower(moving side to side in battles)
XS pokemon generally have less health, take more damage generally also, however their CP is still unaffected by being XS(from what I have noticed), they can dodge a lot faster and have more successful dodges, also some hits from enemy pokemon miss with XS pokemon(my observation)
Pokemon locations reset, if someone got a Dragonite somewhere yesterday, it's highly unlikely you will find more Dragonite in the same place the next day. Don't go chasing waterfalls. Just adventure and you find what you're looking for.(Personal opinion, and it's definitely more fun than way.
Items are very important, you can use incense to attract more Pokemon to you. This helps you catch more, more stardust, more candies, more CP upgrades, more evolutions, great tool.
Same for lure modules, if you go to a pokestop, you will notice a small oblong(not sure if it's an oblong, just go with it) it's above the circle that you spin for items, you can also spin those circles for items if you did know, if you click on the small oblong thing above the circle, you can place a Lure Module, these lure modules benefit everyone who is near that pokestop, so if you want to lure pokemon and maybe some new friends, go for it, people love that pink flowery shit.(also, if you find two pokestops next to each other, sit in the middle, drop two lure modules if you have them, and make it rain pokemon)
Lucky Eggs double your exp, use them with incense and lure modules to get more XP from everything you do, if you evolve pokemon with lucky eggs on, it's 1200XP a pop, so save up your candies and pump an Egg on before you do all your evolutions and watch your level climb.
Potions heal pokemon who have been injured. To use potions, tap on the potion and select the pokemon you want to heal.
Revives work the same way, tap the revive, then revive the pokemon you want to revive.
When catching pokemon you can use razzberries in your bag to make it easier to catch them, and you can switch between pokeballs also in the bag, you start getting better pokeballs (greatballs) from level 12 onwards.
In the pokemon section you can scroll sideways to see how many eggs you have, you can tap on the eggs to incubate them, after you've walked a certain number of kms the eggs will hatch, giving you sometimes very unique and rare pokemon. Keep a watch on your eggs, if you incubate quite a few, pop on a lucky egg and rake in all the exp of all of them hatching at once. If a new pokemon you don't have hatches out of an Egg, it's 3200xp a pop.
With Gym battles, once you win a gym, my recommendation is to assign a good pokemon on there, but nothing super strong, because if a member of your team comes along and they can't beat your pokemon, they won't be able to put on their own pokemon.
Also, it is good to take gyms in groups, once you have taken a gym, keep attacking your pokemon in the gym and beating it to increase the level of the gym, the higher the level, the more Pokemon can be assigned and the longer you can hold that gym.
If you hold gyms with pokemon, for each pokemon you will get 10 coin's every 24 hours(i think it's 24 hours) if they last that long. My suggestion, capture the gyms just before the reset. Saves time and disappointment. You can use the coins in the store to buy more items and upgrades.