"Before I begin handing out advice, I want to say that I am a player who begins at veteran difficulty, and I increase the difficulty after each play-through. So, I have played this game on the hardest difficulty, and have beaten the game on that difficulty. This advice is from my own experience playing, and I will list some general tips."
Leveling your companions:
Like others have said, do not allow the AI to do any of the leveling for you. Do not spec your companions for combat either, as you will be doing most of the killing and fighting. Instead, focus on the utilities your companions have to offer, whether they be crowd control, being able to open stuff, de-buffs. or just being a bullet sponge.
Combat Tips:
When in combat, do not hug the wall to take cover. There is a cover system in Mass Effect 1, but the cover system frankly sucks. You have to go to the very edge of the wall to pop out and shoot, but the enemies can still hit you at the edge no matter what. Instead, it is far more effective to run in and out of cover, like in a First Person Shooter, than actually utilizing the cover system.
Also, pistols are your best general purpose weapon in the game, as pistols, with master marksman, has the highest dps in the game,it is more accurate than the assault rifle, easier heat management, and to me, makes every other weapon outside the sniper rifle, a matter of preference. Sniper rifles are your only long range weapon, which gives them some use over the pistol, but I found there weren't many situations where I needed to snipe. Of course, you could if you want to.
Vendors:
Do not worry about buying anything from the vendors other than your basic capacity upgrades,(but you do not even have to buy those, as grenades and what not are sort of useless)because the loot you find improves as you level up.