Are deore 9 speed shifters compatible with Alivio & Altus derailleurs? I have a 3x9 hybrid with an Alivo RD and Altus FD. The shifters are Altus. Can I upgrade my shifters to Deore M590 without changing my derailleurs?

I plan on changing my derailleurs in the future but don't want to do it all in one go to spread the cost.

I did look at the 105 flatbar shifters but then I would have to change everything and I don't think there is an upgrade "path" short of buying a new bike
Yes. You can take any 7-9 speed Shimano rear derailleur and use it with any Shimano 7-9 rear shifter. And the Deore front shifter will pair fine with the Altus FD.