You've really got to work hard to preserve your squad carefully, early on. Get a hospital and hire some medics in your starting base quickly, and in missions, minimize your casualties. Until you've got a few months under your belt, you will not have enough soldiers in the hire pool to easily replace your losses on Standard difficulty or higher.
Do not let a squaddie go into battle without armor, and give armor a priority when it comes to research. Research long-ranged, high-accuracy weapons (like lasers), as they'll let you engage aliens outside the reach of their plasma weapons. This'll serve you well until the aliens start wielding particle beam weapons, at which point you'll lose much of your range and damage advantage. Until you get lasers, the Machine Gun, Sniper Rifle, and Grenade Launcher will be your constant friends.
Even once you get lasers, don't neglect your grenade launchers. They're exceptional for indirect attacks on aliens behind cover, especially once you learn how to place grenades well - don't neglect the "timed" fire modes!
