First a brief explanation of the terminology. 'Paint cage' this is the tool that you put the actual roller sleeve on. The tube bit that applies the paint is called a 'sleeve' or 'roller sleeve' etc. Sometimes I've used 'roller' or 'paint roller' to describe the cage and the sleeve together.
Loading your roller properly is an important step, the amount of paint you'll want on your roller depends on the surface that you are painting and what sort of roller sleeve you've got but generally the motion is the same. You want to roll the roller down the pan until the just the roller sleeve touches the paint, let it get saturated for a moment before lifting your paint roller up, moving it towards the top of the pan and rolling it back down into the paint. Doing this a few times will load up the paint tray as well and super saturate your roller, you generally want your paint roller to be on the verge of over-saturation as this allows a consistent thickness of paint as well as full paint coverage on the wall. It's also important to try not to smother the entire roller in paint, you only need it on the sleeve so try to keep it there.
Once you've got some paint on the roller and on the tray loading your roller, it's extremely easy and doesn't require a lot of time in the pan. A quick dab of paint on the roller from the pan, lift it back to the top, roll it down twice and you're usually ready to keep on painting.
When applying the paint to the wall the best method is to use long motions going from the top of the wall to the bottom of the wall working in areas 2 to 4 feet wide depending on how tall your wall is and your roller sleeves paint holding capacity. This spreads the paint consistently and gives the best finish.
Once you've got your paint roller loaded with paint, it's time to start painting. When using the paint roller you want to apply just enough pressure to get the paint onto the wall, in most cases and unless you're painting a very rough surface like old brick there shouldn't be any need to overly press or force the roller into the wall this is probably more work than necessary and can leave unsightly roller marks. Start near the middle of the wall, roll the roller up the wall to the top and then come back down to the bottom of the wall re-rolling through the area where you started. Now you should have something like a big straight patch on the wall, on your roll up you want to move in either direction left or right and don't make it too far off, you want part of your roller to still reside in the previous area of paint you applied. On your way back down you continue to move in the direction and now your rolling motion is beginning to take on the shape of a very large V or W. Make sure you go back through your original area of paint to spread the paint out on the wall and give it a consistent thickness. For optimal results, after you've spread the paint onto the wall, finish your patch up by gently rolling downwards over the entire area you've painted, this will give a consistent finish.
Typical problems when rolling walls are over applying the paint, not spreading the paint evenly, under applying the paint, inconsistent finishes and roller lines or marks. Most of these problems are easy to fix if caught before the paint has had a time to dry or set, though with some modern day paints it can be only 5-10 minutes before fixing it is a problem so it's best to check your work as soon as you are finished. If the paint is either over applied and/or not spread evenly the build up of paint can begin to drip or sag, this is usually pretty obvious and easy to fix if caught quickly by just re-rolling the area. If the paint has been under applied/not spread evenly you can see what is referred to as 'holidays', this is typically where an area was rolled only once or twice and you can see that the paint did not fully cover. When roller marks are present this is usually a sign that too much pressure was used when applying the paint, causing it to squeeze out of the roller unnaturally at the edges of the roller sleeve. Most of these problems are easy to detect my looking at the wall from an acute angle. Overall these problems are usually caused by one of the following: poor quality roller sleeves, poor quality roller cage, poor quality/old paint tray that loads the roller poorly or sloppily and simply an in-experienced painter