Changing from a triple ring Shimano Sora chainset to a double ring chainset
You won't need new wheels, but you'll need a new cassette in back. Also, I'm assuming that you've updated your shifters and rear derailleur from 9 to 10 speed? The minimum you will need to switch out going from 9 to 10 speed is the rear shifter, rear derailleur and cassete.

As far as swapping out the triple to a double you don't need a new front derailleur, but you will need to readjust it by bringing in the limit to accommodate the change from 3 rings to 2 so your chain doesn't drop inside when shifting down to the inner ring. You don't absolutely need to swap out the front shifter, but adjustments are a little more tricky as you currently have 3 clicks instead of the 2 needed. You can either limit the clicks to two or just use 2 clicks for shifting down. Be sure to check the outside limit as well. Once the limits are adjusted you'll just need to readjust your front shifting. There are plenty of instructions here and videos online that spell this out completely.

You might also need to adjust the length of your chain, so be sure to check that.