Shimano deraillers are no different between 6, 7, 8, 9 and 10-speed indexing (except for pre-9-speed Dura Ace). 9 and 10-speed chains are narrower. You generally need the same number of cogs as your shifter has speeds for, especially for 8, 9 and 10 (since the cassette is the same width).
If you're going to update, you could just go 10-speed shifters, cassette, and chain. Or 9-speed shifters, cassette and chain.
Or just get new 8-speed shifters if you don't want to update. But your cassette and chain are probably partially worn anyway, so may as well at least to go 9-speed.