What are good champions to ban and what champions/roles are good to first pick when you are team captain? What happens if your team does not want to trade? what then?
Thanks

Pick who you are best with. Don't trade, it rarely works right. The common bans are usually blitz amumu malph and shen. After that it's sort of personal preference.

I tend to ban junglers that will carry a game. Vi and Hecarim are good bans if your jungler isn't going to play them because they can be game breakers.

If you have absolute first pick, then you'll want to see if you or anyone wants one of the common bans (Amumu, Malph, Blitz, Shen) and then trade with them (you have to make this happen fast, if your team isn't responsive just abort this idea). Make the other team waste their bans while you make legitimate, strategic bans.

If team captain, ask if your team has and preferred bans first of all, and ban those. Then come the basic bans.

If you're in low Divison (Bronze, Silver, Gold), bans usually consists of Amumu, Shen, Blitzcrank, Malphite, and two others, usaully Hecarim/Vi/Darius/Akali).

In higher divisions (Plat+) bans are usually TF, Thresh (more worth than Blitz IMO), Shen, Zed, the newest champion (i.e. currently Lissandra), and usually a top laner (Elise/Zed/Zac).

If you're first pick also you can ask if you want to trade someone, since usually first-picking a mid lane can be hard. Also if a champion that is highly contested is open, ask if anyone wants it.