Does your first champ pick matter (like jungle or mid)? Is it mostly counter-picking, or do you still choose a team that your team has already decided?
I played a lot of ranked, almost 1600 elo atm so I'll try to answer. In order of importance, here is what I think goes into the picks:
Each person on your team should have a role. You don't want to pick multiple champions that can only jungle or only mid/top. Unless you've agreed with your team pregame, usually you have 2 bottom, 1 jungle, 1 mid, 1 top. Pick champions for available roles.
In a lot of organized teams and high elo games, people usually pick ad carries or support champions first, since these are generally hard to counter because of the role they play. The ad farms and the support provides utility, roles that's difficult to counter. Mind you, there are specifics like sivir being good against Caitlin.
For the rest of your lanes, you have to be very careful about counterpicking. This is why it is always a bad idea to first pick kassadin, as he has numerous counters that simply prevent him from laning at all.
A lot of people also pick champions that can fill multiple roles to throw people off. For example, when I pick gangplank, he can go top or jungle. If my opponents pick pantheon, gp's hard counter, I go jungle and have a teammate pick another top.
You have to pay attention to how your team can work together and work to win team fights. Picking for roles and counters do little if your team doesnt work. I'll list the most common team comps:
A. Poke team Your team has the ability to do significant damage from a distance over time. This allows you to slowly push down towers or seize objectives.
B. Standard team with moderate CC Basically you have some crowd control abilities that allow you to protect your carries and shut down the enemy team.
C. Gank team Your team has a lot of mobility and always maintains map presence. Good examples of gank teams include heavy ganking junglers like alistar or champions that can pressure globally like shen, karthus, or twisted fate.
There are many more, experiment!
Normally taking AP or Top champions 1st is a bad choice because its all about countering your opponent's lane and if you take those 1st they will normally counterpick you.
Thats why mosty supports/junglers are picked first because it doesn't matter much (at least in higher elo). Jungle speed does matter but you can't counter everything.
It depends a lot on what champion you are picking for a specific role. There are champions for each role which are considered "safe picks" because they aren't really hard countered by any other champion. Examples would be: Top: Irelia, Kennen, Riven Mid: Ryze, kennen, cassiopeia Bottom: I mostly just play Aps and sometimes top in ranked so I cannot really comment much on bottom lane, also there are way more variables there since it is a combination of 2 champions.
Another thing to consider during picking phase is to see what the opponent is banning as it often indicates on what they're going to pick on a specific lane, for instance if someone bans ryze and leblanc, there's a good chance they want to take ahri since those are both counters against her.
Hopefully that helped you out a bit.