Whether someone is moving to the apartment next door or clear across the country, a mattress is one of the heaviest and most cumbersome items to move. Many people have damaged their mattresses trying to move them, or even worse, have strained their back or knees in the process. Damage and injury can be avoided if the mattress is carried properly.
Always carry the mattress by the side (i.e., so it's not flat). Carrying the mattress in this way avoids putting stress on the center of the mattress, which is intended always to have support underneath it. It also makes it less likely that you will damage or get dirty a large portion of the mattress if you happen to drop it. Try to get the mattress as perpendicular to the ground as you can. Otherwise, the weight of the mattress will want to pull the mattress down into a flat position.
Always ask a friend to help you carry your mattress--you can take one end while they take the other. Mattresses, even ones that are not excessively heavy, are awkward to heft alone, and if you try to do so, you may end up losing your balance and getting hurt in a fall. Even in the event that you manage to hoist the mattress, it's hard to see where you are going, and if you drop it, you'll probably damage it.
Try using a mattress tote. These have many designs depending on the manufacturer, but the basic premise is that straps slip underneath the bottom side of the mattress and angle up to the edge where you can hold onto the straps well. You then heft the mattress using the straps rather than just your hands. This is helpful because it prevents your grip from slipping, and because the straps, which start in the center, put support where ideally a third person would be providing it. The carriers also stop you from having to lift the mattress very high and eliminate the need for deep bending.
Place a protective barrier over the mattress as you move it. You can buy an inexpensive plastic protective cover, or you can just toss a sheet over the sides. Either will protect the mattress from dirt and damage in case it is dropped.
If you do not have a carrier, bend your knees slightly (do not bend at the waist) to prepare yourself for the lift. Then grab and lift at the center rather than the bottom. This will give you far better control over the mattress once it is off the ground than carrying it from the bottom. It also prevents you from bending too far and lifting the weight of the mattress with your back.