I've been wondering, what could be the best ways for finding and capturing pokemon in the games
Just a hint or two.

Always have plenty of Pokéballs at hand, you never know what is going to come up and depending on their capture rate, you will be needing them.

Yeah, you could run in to a shiny!
If it's night or you're in a cave have some Dusk Balls handy - They work better than ultra balls in the right conditions.
Have a team of all levels or a level 100 with False Swipe, and a Pokémon that knows a Paralysing/Freezing/Sleeping move.

what works for me is the pokeball on the bottom screen as soon as the pokeball you threw closes you press right in the center.

I've caught over 50 pokemon like that first shot.
(I DON'T HAVE THAT MUCH LUCK)

Coincidence.
You can't change a Pokéball's catch rate. Period.

I wrote up an article on catching runners in Gold, Silver, Crystal, Ruby, Sapphire, Emerald, FireRed, LeafGreen, Diamond, Pearl, Platinum, HeartGold, and SoulSilver:
http://floatzel.net/pokemon/runners.htm

As for everything else, collect a great variety of balls. Use each ball during its relevant usage time.

The most useful Balls you'll be needing on capture expeditions:

Any type of water ball (i.e Dive or Net Ball): Has a higher chance of catching Water Type Pokemon
Timer Ball: Catch rate increases depending on the length of the battle (Useful for legends)
Quick Ball: Catch rate increases depending on how early the battle is and decreases every four turns (somewhat useful on roamers)
Dusk Ball: Higher catch rate in dark areas, like nighttime and caves (useful for any occasion)
Ultra Ball: If all else fails, use these.
Master Ball: If you're really pissed off at a roamer, use this.

I'd recommend dusk balls for everything, if your trying to specifically catch something in particular that is. Reset the clock to night if you have too.
But stalk up on balls of other types, especially net balls, nest balls, and quick balls. If you are attempting to get rare candys via the pickup ability, then you will also have ultra balls on hand, as you will find many....
Aside from what balls to use I would recommend carrying around a lower leveled HM slave or two, simply so as to be able to drain away some HP on the fly. Also including a breloom, or smeargle with spore for sleep is a good idea. If you are actually going hunting for a pokemon, then bring false swipe.

Thanks people. I'll use all your tips for my advantage. Hooray!

MASH THE A BUTTON AS FAST AS YOU CAN!11
...
Is what I use to think as a little kid xD

I always carry around my breloom in the lead since it noes spore. Status ailments help a ton for catching pokemon.

EDIT: just read serenades post. I was beaten to the punch with the breloom thing. ;w;

I like my level 100 Garchomp with False Swipe.
Too bad I lost it when I got 'Corrupt Data'. Pfft.