Upgrading from 9-speed Tiagra to 10-speed Ultegra piece-by-piece will be difficult because the number of speeds of the rear shifter and cassette must match, the chain and FD should match, and the chain must be at least as many speeds as the cassette.

There is an exception to the above. You could replace just the shifters for now and use Sheldon Brown's alternate cable rating to mate them to the 9-speed cassette. Then later you can replace the chain, cassette, and FD. May as well run the cassette and chain into the ground, so it would be time to replace those anyway. The RD could be replaced at any time you'd like. Edit, forgot the crank, which should match the chain, but you might be able to get away with mis-matching them.