Vanguard has the steepest learning curve and difficulty curve in the game.
The first thing is that as a new character on Insanity you will not be able to charge at will. You do not have enough upgrades and talents leveled up to be durable enough. What you can do is attempt to charge enemies at the edge of their lines or lone stragglers so that you do not get overwhelmed.
Bonus power choice can also make your life a lot easier or harder. I prefer to take Slam for new Vanguard. Partly because on PC it won't let you select Stasis for some reason, and partly because it is one of the "1 point wonder" powers. It is especially useful for The Professor due to the number of enemies on ledges and the fact that during the lift phase it sets up warp explosions.
Since I also do Mordin's mission first typically, my squad will be Miranda and Jacob. I try to focus on Pull first when leveling Jacob, and for Miranda Warp is a little better for this mission than Overload. Set Jacob to Pistol, and leave Miranda on SMG during the Blue Suns portion, then go to Pistol after you get into Blood Pack territory (because it is all armor). You should pick your weapon depending on the target, but you will likely end up using the Pistol mainly due to how much armor there is in this mission.
Basically what you will want to do is strip armor with pistol, then use either your Slam or Jacob's pull once the armor is stripped from a mook, then hit that one with a Warp to cause an explosion (if there are other nearby enemies). For targets on ledges, you can potentially pull them to their death, so strip their armor with weapons or Warp then use Pull.
In any case, once you get some shield and health upgrades the game will get easier, but that is a few missions away. Redundant Field, Hard Shields, etc all help. It basically just takes some practice.