Yes, upgrades are also very important, most importantly those you receive by talking to members or that get added when you complete their loyalty mission. Most important are Jacob's ship armor, Garrus' thanix cannon and Tali's deflector shields. Just for safety reasons, upgrade everything, leave no upgrade un-researched. Use the trainer if you need to to get materials or probe the planets either way is fine.
Don't rush to finish the missions, take it nice and slow, check everything there is to check. I've read that if you go without loyalties and upgrades you get the "Bloodbath" achievement. (jk)