Can I just fit a cassette with more gears with the same derailleur?
If you want to change your cassette to something with more gears, then you will need to replace the rear shifter and possibly the rear derailleur.  A cheaper option would be to get a wider range cassette which still has 8 gears.  You'll have bigger jumps between the gears, but you'll have a bigger range of gears.  With a wider range cassette, you may need a new rear derailleur with a longer cage.