When I first started the game, I built an Infiltrator setup, using cloak and sniper rifle.
After a while I got bored though and switched to a Vanguard build that is both effective and fun to play:
Weapons: Asari Sword, Pistol (go for a light sidearm. Predator works fine), Assault rifle, Shotgun. Make sure you are under 100% weight limit, for you need short cooldown time to make this build work well.
Skills: Biotic Charge, Nova, Pull
Gameplay: Strip shields with assault rifle, prime target with pull, trigger biotic combo with charge. Then finish off whats left with a shotgun, mele attack or a nova.
Biotic charge also recharges your shield so you have to spend less time hudding in cover waiting for your shield regeneration to finaly kick in.
Pull also works well as a "leave me alone"-button when an unshielded trash mob flanks you. Just suspend it in the air and you can concentrate on doing something else.
Companions: Cora works well thanks to her shield strengthening skill. Also her agressive gameplay style (she is a Vanguard after all) tends to draw attention away from you. Second companion isn't that important imho. Jaal, Vetra or Peebee worked quite well for me sofar.
It is a very fast-paced, dynamic play style. You will spend relatively little time in cover and more zooming about with biotic charge und jump.
You go right up in an enemy's face, but only if you know you can either finish it off quickly or have a plan for a quick retreat.
I found it much more challenging but also more fun to play then sitting in cover at the far end of the battlefield taking potshots at heads popping up from cover.
Biggest problem this build seems to have are snipers at long range and enemys you can't kill in one attack.