Wild pokemons are all over the place. I believe there is some sort of time based setting where if you leave your phone for long enough, a wild pokemon is going to spawn near you regardless of where you are and regardless of what the "nearby" button says.
On the map, if you see areas where you get wild grass animating, pokemon are more likely to be around those areas. You will normally find multiple pokemon in those areas and your more likely to get multiple to spawn at the same time. Though i believe everywhere can spawn a pokemon potentially based on a timer since your last encounter with a wild pokemon.
If you see a wild pokemon on the map, you want to try and double tap on it - this can be painful if they are right next to a pokestop/gym - as the pokestop/gym take preference and the app seems to "help" you select these rather then the pokemon.
Double clicking on the pokemon will take you to the capture screen where you attemp the capture.
Nearby Pokemon is actually quite powerful once you know how it works. It basically seems to be a linear scale in regards to the distance to the Pokemon and the number of footsteps. The only one that i think may not be this way is the 3 footsteps. I think the 3 footsteps means 3+.
I added an image that shows the distances based on a test i've done. 100% of the time that the Pokemon has had 0 footsteps, when i have clicked it, it has spawned within 5 to 10 seconds. I have also 100% reproduced that once a Pokemon goes outside your circle range, it changes from 0 footsteps to 1 footstep
Image That Shows Footstep Distance
Once you are in the capture screen, the goal is to throw the pokeball so that it hits the pokemon on the full. If it bounces, goes over them or misses, well you have to try again. Each throw will take 1 pokeball so you dont want to miss. In saying that, you will eventually get your eye in where you can hit them with 1 shot 80% of the time.
To throw the pokeball, tap and hold the pokeball, then flick your finger with the right speed/power based on the distance the pokemon is from you.
The color of the circle represents the difficulty of the capture. Green being the easiest, red the hardest. A harder pokemon is more likely to resist capture and escape even after you successfully hit it with a pokeball.
After much testing, i believe the entire purpose of the colored circle is to tell you the difficulty of the catch and it also highlights the area you need to hit to get "Nice" and "Great" throws. Pokemon can still escape capture even if you do a "Great" throw.
I have not seen a difference in capturing chance when the colored circle is small or large
I have not seen a difference in "throw speed" if the circle is small or large.
So i'm unsure if the below is true based on my own experience - left it here as its from official site
* From the official site, you have a greater chance to capture the pokemon when the circle is the smallest in diameter. /u/PatThePirate
You gain some extra experience depending on the type of throw you do and whether you hit that colored circle.
You only get 1 bonus per throw, so even if you do a curve ball and get a "great throw" , only the highest XP will show - so you will get 50xp and not 60xp.
There is a random chance that the pokemon resists capture. Most of the time, they will stay around and you simply need to throw another pokeball at them to capture them. However, sometimes they will run away and they disappear from the map and you have missed you chance to capture them.
I have taken up to 5 captures to actually capture a pokemon before - mostly, if they escape capture once, the next time you get them with a pokeball, it will work. Its rare that it goes more then 2 times.
To do a curveball, you want to tap and hold the pokeball, rub in a circle so that the pokeball rotates, then flick it like you normally would (except its going to curve obviously). If you manage to hit a pokemon with a curveball, you gain an extra 10xp (you normally get 100xp from a capture - 600xp if its a pokemon you dont have yet).
I have had a pokemon escape capture after being hit by a curveball - i'm unsure if it effects capture chance or not, but its definitely not 100% chance to capture if you hit with a curveball.
But curveballs are quite fun.
Once you get up a few levels, you will get some berries, you can use these berries to make your next throw have a higher chance to capture the pokemon. But it only effects the next 1 throw (regardless of if you hit or not).
Berries dont make it a 100% chance to capture, but i definitely feel a difference when i use them on harder Pokemon and it really feels like they dont escape hardly at all - but i have had it happen to me now once or twice.
As you level up, you get stronger pokeballs which make capturing easier as well.