To begin, make sure you are wearing a neck strap. It is very difficult to hold a saxophone without one. Whatever type you should choose, from a padded neck version, to a harness to a no frills hard cord, the hook should be somewhere in the middle of your chest.
It is important not to purchase or use a neck strap with a clip like the one in the image shown here. These kinds of clips, while they are easy to attach to your horn, are not strong enough to keep the saxophone from falling. What happens is that the weight of the horn pushes the clip off of the ring on the back of the saxophone and you end up dropping your sax. This is bad. Spend the extra couple of dollars to get a clip that will close permanently like the one shown below.
The first thing to realize is that there is no left and right handed saxophones. All saxophones are held the same way. Your left had goes on top and your right hand goes on the bottom. This part is simple.
The location of your fingers is also simple, the first finger of your left hand should be placed at the top of the “stack” on the first key. Same goes with your right hand. It should be at the top of the bottom stack. See the attached images for an example.
There are two ways to place your fingers. You can either place them flat on the keys, using the divit on the finger to contact the key, or you can curl your fingers like a guitar player does and touch the keys with the very end of your finger. I spent some time in college looking into which was the better way to do this. I think there is a difference. Not everyone might agree with me but I do believe that you can play faster, with more control, if you curl your fingers and use the tips to contact the keys. I discovered this when I was seeking information about Charlie Parker from the 
library at College. They had a video library and I took out a video of some of the only video footage of Charlie Parker playing. It was more than clear to me he is not using straight fingers, he is most certainly curling his fingers and contacting the keys with the tips. If it works for Charlie Parker, the most famous of the Bebop saxophonists, someone who was capable of lightening quick finger control, maybe there is something to it and we should all be doing it. In my playing I tend to alternate between both straight fingers and curled, but I seem to find myself reverting to curled when I want to play quickly with control. Do what feels comfortable for you and what position you can achieve without hitting keys you do not want to hit.
It is quite easy to hold the saxophone. It is hard to hold it properly. What I mean by this is that it is very important to remember one simple thing, never move your body to meet the saxophone, always bring the saxophone to your body.
Start by sitting comfortably in a chair with no arms. Don’t lounge back in it, sit up straight at the end of the chair seat. Position your horn comfortably without moving your body. Tighten or loosen your neck strap, move the neck left or right, place the horn to the side or in front. Do whatever it takes to bring the horn to you, do not go to the horn. This will make it easier to breathe and to play in general. Remember, when you are in a car, you do not go to the steering wheel, you tilt it, pull the seat up, forward, whatever, to make it comfortable to drive. Use the same concept here, adjust till you are comfortable and are ready to take a long drive. This will make it easier to play without a doubt.