Guides and Basic Info
Here are multiple websites and forums gathered to give you some knowledge on the game. (Credit to @Swim for compiling them)
How do I catch Pokemon?
You will be swiping your finger from bottom to top and releasing your finger when you feel the distance is enough. Releasing your finger earlier will make the Pokeball not fly as far as releasing later. Different sized Pokemon will be at different distances so not every swipe will be the same, smaller Pokemon will usually need short swipes, larger Pokemon will usually need longer swipes.
A circle will appear around majority of the Pokemon's body and another circle will constantly shrink until it's nothing and will reappear again beginning as large as the white circle. The aim is to throw the Pokeball and land within the shrinking circle. Sometimes the circle may not be centered around the Pokemon's body for example Onix's circle will center around it's head and the head sways curving left to right. It's best to keep your finger on the Pokeball first as if you were going to throw, bit watch the animation instead to get the pattern down before throwing the Pokeball at a new Pokemon. If you do manage to hit within the circle you'll get words 'Nice (large circle), 'Great!' (medium circle), or 'Excellent' (tiny circle) which means you'll have a better rate of a successful catch. The smaller the circle the greater of an EXP bonus you will receive as well Nice nets you 25 EXP, Great 50 EXP, & Excellent 100 EXP.
At random the Pokemon will either jump up or do an attack animation. Jumping will still allow you to catch the Pokemon if you throw it right, their attack animation will make them invincible to Pokeballs for a short amount of time.
Remember this is mostly an RNG game so even if you do get an 'Excellent' throw it's not guaranteed that the Pokemon won't break free.
Are there any ways to make catching easier?
Yes, when you start Leveling Up more items will become available to you that you can use to make the capture rate higher in your favor.
First off would be Razz Berries, which become available when you reach Level 8 both gifted and from PokeStops. Simply go to your Item menu while in Capture mode and select the Razz Berry. Tap it's image when it's hovering over the Pokemon you want to catch and it will make the next throw have a better capture rate. Make sure to actually hit the Pokemon with the ball because you only get one chance at using it, which will turn into a wasted berry.
What is the easiest method of adding my own Pokemon into a Gym that already has some of my team's Pokemon in it?
As long as the gym isn't capped at Level 10 then you will have the opportunity to add your own Pokemon in by training. Hopefully you have a Pokemon that is able to at least defeat the first Pokemon at the Gym because that is essential. Battle the Pokemon defeat it and as soon as you get to the Pokemon that you can't defeat either due to the CP of it being too high or you are running low on HP use the 'Run Away' button, bottom right of your screen that looks like a Pedestrian symbol as quickly as possible. Heal up your Pokemon and rinse and repeat until the Level of the gym gets high enough where you can add your own Pokemon into the gym. Note that the Gym will gain more Prestige/HP if your Pokemon that you are training with is lower or almost equivalent to the CP of the Pokemon you are defeating. The higher the Gym's Level is the more Prestige/HP you will need to add in order to make room for your Pokemon. Hopefully the Pokemon that you choose to leave at the Gym isn't lower than what you've been defeating because once rival Gym members try to take it over it'll be the first to go.
What coordinated groups do is put a lower CP Pokemon as the first Pokemon and they continuously defeat that one Pokemon to raise the Level/HP of the gym to very high levels before adding their Pokemon. Usually they'll get it to Level 7 or so, since groups usually consist of just a bunch of friends who can fit into one or two cars.
Just hope that you are the only one in the area while you're training because other team members can and probably will take the open slot if it becomes available. And if you notice the Prestige/HP of your Gym going down you may as well stop because other teams are battling at the gym and it is much easier to take the Prestige/HP of the Gym down than to build it up.
ARE THE SERVERS DOWN?!?!?!
You can check these links here!
*None of these are official and I believe most are user-based
When Pokemon Go was announced!
They show barely anything game play wise, but it's for Android and iPhone so no one can really complain.
Some people may end up traversing the real world though, the horror