The rear derailleur is the cage like thing with 2 pulleys that makes the chain move from cog to cog. Changing that will do nothing to your gear ratios.
What you appear to be asking about is swapping out the cassette (the stack of cogs on the rear wheel) to a wider range. Depending on what you have on there now, you may already have about as wide a range as is available in a cassette. Seeing as it a MTB, this is probably so. In that case, if you don't have enough gear range you will need more than one chain-ring on the front to get the range.
To be honest, I don't think this is really worth putting any money into. Just leave the front in the middle ring and shift two gears at a time for the rear. I think that as you ride more you may find yourself wanting those closer steps in the shift pattern. I know that on my road bike with a much narrower cassette range, there's still a couple of steps I'd like to tighten up if I could only get strong enough to ditch the larger cogs.