Installing the Seat
Do this in advance. You don't want to have to figure it out in the hospital parking lot.
According to the NHTSA, the safest position in the car is whichever rear position allows the seat to be installed as tightly as possible and at the right angle. Generally, the middle of the rear seat is considered safest, but if there is a indentation or bump in that position, or some other factor that makes it difficult to install the seat properly, then move the seat next to a window.
Your main objective is to get the seat in tightly. Push down on it with your full weight with one hand while tightening the seat belt with the other. Lock the seat belt according to the vehicle's owner's manual instructions. The seat shouldn't jiggle more than an inch in any direction. If it does, check to see whether the buckle is fastened right at the place where the belt turns to go through the slot in the safety seat. In this position, the belt may be able to loosen. Unfasten the buckle, twist the shorter end of the belt once, and reattach it. This will keep it tightly locked in most vehicles.
If your seat belt has a free-sliding latchplate, one through which the belt can slide even when it's buckled, you'll need a locking clip. All new car seats come with one. If you lose it, you can buy a replacement from the manufacturer, or from retail stores such as Babies R Us. Here's the right way to use it: 1) Install the seat tightly; 2) hold the two parts of the belt together at the latchplate and unbuckle it; 3) thread the belt through the locking clip, no more than one inch from the latchplate; 4) buckle the belt again.
Harness straps on a rear-facing seat should come through the slots that are at or slightly below your baby's shoulders.