As has been said, the N260 is a stunning road on the Spanish side and the road between Sort and Vielha (C13/C28) is amazing. Another good and often overlooked one on the Spanish side is the road between Guardiola and Campdevanol (B401/402) - the C26 between Ripoll and Berga tends to be ridden more often. The N135 from Pamplona to St Jean Pied de Port is a cracking way to cross the mountains at the western end; going through Andorra is well worth it in the east. Any of the passes in the middle is good!
The French side has the higher passes and, for me, the more demanding/more rewarding riding. There are plenty of good, smaller roads that are narrower and a bit rougher than the main Spanish ones.
You can work your way west-east along the French side, stopping at St Jean Pied de Port, Lourdes, Foix, then finally stopping in Andora, enjoying great riding all the way - especially if you're criss-crossing from the French side to the Spanish side and exploring all the little by-ways.
Look out for the Tour de France cols - Aubisque, Soulor, Tourmalet and Aspin - near Lourdes. Near Foix, stop at the ruined castles of Montsegur and Peyrepertuse (take walking boots - you have to ankle it up from the parking spaces, but it's worth it). The citadel of Carcassonne is definitely worth a visit as well, though the city is obviously busy.