LuptonM
There are various different ways to make any item, which is why they come in all sorts of shapes and styles.
Rob's suggestion of getting a manual is spot on. You really should read up on the different joints and techniques, and look at some project plans to see various ways to do this. Ultimately, the answer is going to be - whatever works for you.
However, in simple terms, a chest of drawers is a big box. And there are two main methods of construction for anything like that. Slab, and rail-and-stile.
Slab means that you directly join the 'slabs' of wood together. For that, the simplest way would be to drill holes in both pieces and glue in dowels. Altneratively, you could cut mortices in one piece and tenons in the other, or you could even decide to use box or dovetails joints. Even simpler - you could just glue the pieces together and use plastic doo-das to strengthen the joint inside...that would be the 70's prefab method which I would not recommend. But you get the point. There are different ways.
Rail-and-stile is also called frame and panel, which probably describes it better. With this style, you would be making a frame out of square-section wood - eg.32x32mm to create a decent, solid framework which forms the edges. The panels would then usually fit into grooves (rebates) you cut into the edge pieces, which also allows for movement. Or they could be dowelled in.
If you look at the sort of things you get from 'Pineworld' type places , most of it is slab, which is cheaper and quicker to make. Personally I would say slightly less attractive as well.
The chances are that you would mix various methods. You might use rail and style to make the main section, but dowels to secure the lid, for example.
For the runners, there are probably hundreds of possibilities. Personally I would just glue and screw, but for a stronger joint you could also cut a rebate into the side frame. There are also runners you can buy and screw on. It would depend on whether you are making something to house a rock collection or lace underwear.
The back is often just made of a cheaper option - like tongue and groove, thin ply or even hardboard. So the main strength comes from the sides, top and front rails.
To get the panels straight, I assume you mean you are planning to use MDF or ply. In that case, you just need a method of sawing - a panel saw, or a circular saw or even a table saw - and a method of ensuring your line is at 90 degrees. For that you will need a thing called a try square. You could also buy planed boards and glue them together to make the panels. In that case, the edges should already be square.
Finally, if you google free plans, you should be able to find something to help.
I'm fairly new to this myself, so apologies to anyone if I got any of my terms wrong or left stuff out - but I know how confusing it can be when you start trying to understand how stuff is made.