You can just buy new shifters in theory, but there isn't much 8spd around for road anymore. So, you may need to replace most of the drivetrain to be compatible with 9 or 10spd. It is possible to block out certain gears with shifters that work for more gears than you have, but it's not really a good long-term option.

You'll probably need a new rear derailleur, new cassette (possibly new freehub on your wheel, but that depends), and new chain. The crankset and front derailleur may or may not be compatible. However, if you're changing everything else, you might want to look into a full upgrade kit (There are various configurations). With that kit you are likely to get a volume discount on the parts, which is cheaper than getting most of the components piecemeal.