What type of screws should I use?
Depending how you make your joints, if you choose to use screws, they need to go into the joining piece at least 3/4"-1" deep, and deeper when going into the end of a piece, 1 1/2" is a good depth, For this you can use a 2" screw, countersunk a little.
The drawing below, may have a little more info than what you are looking for, but it will give you options if you choose.


I would suggest putting a 1/4" thick plywood back, thinner will work too if available. I have cut this with a razor knife if you do not have a table saw. The back can be nailed on with 1' brads after the shelves are assembled



If you are not concerned about the looks and it will be a utility type shelf these are the screws I would use. Still nail the back on with 1" brads


Any of these countersinks will do what you need, make sure you keep your drill perpendicular to the wood and in the center of the shelf for the thickness, about 3/4" from the edge from the front and back.