For starters, ask your Tryndamere player if he can learn to play Soraka/Janna/Sona/Taric. Soraka and Sona are easy to play and add much more value to the team. Once he knows those two he should learn Janna and Taric. (Ok it's not ideal to tell him to switch his main, but for teams, this will be a lot better).
Then for other picks: if you don't know which champions are strong and commonly picked, watch (tournament)streams. This will give you an idea of which champions are popular. However, never blindly copy their setups, instead be on voice chat with your team and discuss why they picked X and not Y. Do the same with jungle routes and map objectives.
And, most importantly, learn by playing.
For starts, here is a list of commonly picked champions (in no specific order):
Jungle: Amumu, Nunu, Rammus, Trundle, Warwick, Nocturne
Mage: Anivia, Malzahar, Karthus, Ryze, Rumble, Swain, Kassadin, Twisted Fate, Alistar
Ranged AD: Ashe, Fortune, Corki, Caitlyn, (Vayne).
Bruisers: Irelia, Renekton, Jarvan
Tanks: Malphite, Shen, Gragas
Support: Janna, Soraka, Taric, Sona
Teams USUALLY consists of 1 Support, 1 Jungler {tanking junglers}, 1 Mage, 1 Ranged AD and 1 Mage {usually a mage who can take a beating, like Ryze or Swain}/Tank/Bruiser (depending on the situation).