Shimano are the biggest player in the gear industry. Other good brands out there are Sram and Campag. Campag are better known for their road groupsets. Sram operates on a different ratio of cable pull, so compatibility is a bit of an issue.
With regards to upgrades you can upgrade your gears and still keep a Shimano system. It's easiest. By the sounds of it you have a freewheel system. This means your rear cogs have the freewheeling mechanism in them, and the whole assembly screws on to the rear hub.
The "upgrade" is to change to a cassette style hub. Meaning the freewheeling bit is contained in a sealed "freehub". These last longer, and are replaceable when they wear out.
Generally these are 7spd and above, and freewheels are 7spd and below. If you upgrade though you are looking at changing the shifter, rear hub, cassette and chain. More than likely the chainrings also. It is a pricy upgrade, but will offer you a slightly better gear range, smoother + quicker shifting, and a lighter system. One downside is that narrower chains (as found on 8,9+10spd systems) do tend to wear quicker.
This is one reason why I am a fan of the Shimano hub gears, they have the benefits of a wide gear range, but running a thicker, single speed chain they last much longer.
If you are looking to upgrade a derallieur system though, you can't go far wrong with Shimano deore 9spd. It tends to be the benchmark for quality derallieur systems. If you want to really spend some money, it is worth looking at Shimano XT equiptment. The Sram equivalent is x7 and x9, the main difference being that Sram equiptment feels a bit more "chunky" I find. The gear shift has a bit more of an industrial feel to it(due to the 1:1 cable pull ratio). Some like it, some hate it. A bit like Marmite I guess.Only Marmite won't do your gears any good.