The best thing you can do to upgrade or improve any bike is the tyres, this will make the biggest difference to your cycle, making it an easier and faster ride.
I see a lot of people cycling around with tyres not at the right pressure, I also used to do this as a kid and its only now I can see the error of my ways. Put some air in your tyres and not only that, put the right pressure in. Most tyres will give you a PSI range on the side:
In the example it shows a range of 60-80 PSI, I normally go for the higher end of the range, you can really feel the difference!
If you are cycling on the road then you need less grip and thinner tyres in general. If you are offroad then more grip and thicker tyres is normally the answer. People who have a hybrid bike or do a bit of both then there are many tyres that fit inbetween to give you the best experience. You can purchase tyres from sites like wiggle or sometimes the best bet if your not quite sure is to go to your local bike shop, tell them what you are using it for and your budget then they will recommend something that works.
Next we have the wheels, they consists of a hub in the middle, some spokes and a rim a similar size to your tyre. When I first purchased my hybrid bike the wheels were reasonably cheap and I had a lot of problems with them going wonky. Normally this is because some of the spokes have gone a bit loose. The spokes all need to be at the same tension and a spoke key can be used to tighten or loosen spokes but this is very fiddly balancing them to the right tension. I found it was worth spending a bit of money on a decent set of wheels to keep you on the straight and narrow.
MORE TO COME