Check that the roller has a heavy wire frame and that you’ll be able to get replacement sleeves (the bit you apply paint with), so you don’t have to bin the whole roller when the sleeve gets worn or you fail to clean it properly.
Roller sleeves take a great deal of cleaning (in water) and if you inadvertently leave any paint to dry on the sleeve, it may ruin the finish the next time you paint. If you don’t have time to clean the sleeve properly, it’s tempting to leave it to soak, but this can cause the frame to rust, which bleeds into the paint when you use the roller. So take the sleeve off the frame first.
The roller should be stiff, otherwise it will bend when you apply pressure, causing you to paint unevenly. To stop a roller shedding fibres, wrap masking tape around the sleeve - when you peel off the tape, any loose fibres should come away with it.
The fibres (known as the nap) are made of different materials and come in different lengths. Some are designed to produce a smooth finish on interior walls; others are for rougher surfaces, such as exterior render. Generally, the shorter the nap, the finer the finish will be.
It’s handy to have a roller with a screw-thread end because this allows you to screw an extension pole into it for painting ceilings and really high walls, although you can get extension poles/rollers in one. For painting behind radiators, use a radiator roller, which has a small head and long handle so you can reach down behind the radiator. You’ll have to remove column radiators and heated towel rails to paint behind them - summer’s the ideal time to do this job because you don’t need the heating on.
One of the main problems with rollers is that they leave ‘tracks’, which are lines of paint from the side of the roller that ruin the finish. The more you try and roller these out, the more they can appear, which is very frustrating. Rollers with bevelled edges are less likely to leave tracks, so look out for these and always check the surface you’ve just painted for any tracks you’ve missed - it’s often easier to remove them lightly with a paintbrush.