Besides banning champions? If you pick your favorite champion, you get counterpicked and then your team rages. If you pick a champ that isn't a high priority to counter, then you might not be as good with it as your favorite/main.
I once had a guy ban 3 top laners, never responded to the team in champion select as we tried to discuss bans, didn't say a word, and then immediately picked Nasus.
Don't be that guy.
Nobody like that guy.
Idk. If I'm first pick, I just ban whoever is FOTM or a jungler that i hate (lee sin).
Then i proceed to pick lux or anivia, depending on what i feel like doing. Working well so far.
If you are first pick its usually best to just pick a comfort pick, for me its Ahri, and if I get countered, Oh well. Many times in my experience people who pick counters are often times just doing that, picking what they perceive to be a counter, this is especially true in Lower (my) divisions/elo's. Many times player A will pick to counter player B, when they Player A has 5000 games on their champion and player B has like 2. Usually player A will win. Its about knowing your champion and about knowing the match up. There are VERY FEW true perfect counter picks in this game :)
You could also jungle or pick a champion that can go in multiple lanes.
That's a difficult thing to deal with. I ban stuff that I know is globally bad to be on a team, so typically I'll axe a jungler like, Rek'sai who despite multiple nerfs just applies too much pressure for most junglers to keep up with, a super strong roamy mid laner like Ahri, or Katarina, and either a flex pick, or a super annoying champion. Flex being something like Kassadin, who can effectively be top or mid. Annoying being something like Blitzcrank, because nobody seems to know how to sidestep a hook and getting hooked is typically death all the time.
In terms of picking, I pick something that I feel exceedingly confident in, such as Talon, Zed, LeBlanc. If you've played it a lot, and understand the workings of the champion, as well as the basics of what you're going against, you should at least do okay. Not getting nervous about a counter and knowing why it counters you are all you need to play against it. If it's really going that bad you can see if you can swap lanes.
I always try to have at least 3-4 champions in my role of choice that I'm strong with, that way even if you're not in charge of bans, you can ask to have a specific champion left open and you'll typically at least get a single champion you're strong with.
Another thing, which is extremely risky, is you and another player can cooperate, and you can potentially pick for him as first pick, and have him select yours when you see what the enemy team is likely up against. This depends on the person not wanting your role, actually cooperating with you, and in general having teammates that talk, which seems to be a rare luxury in lower elos.
I always pick my main, I don't care if they build a whole team comp around countering me, I can usually lose lane and win game because I know how to play from behind on my favorite champ.
I pick whatever i felt like playing. The only thing I might not do, is pick a champ who really gets dominated by things, like first picking Ashe or something. Thats just because I dont want to face leona/j4/vi/etc levels of combos to where I get shut down. I could care less about getting countered in lane matchups. I'll first pick that Nasus without a care in the world.
i usually take a mobile ADC or a top/mid flex pick if im first picking.
I usually just pick comfort, even if i'm behind on a champion I know really well I can still make something happen ^.^
