I haven't traversed the whole range, but I've toured in the Pyrenees twice and crossed several of the high passes, including the Col du Tourmalet and Port d'Envalira (before the tunnel existed). Obviously, you need to be in decent cycling shape. Apart from that, I didn't do any special training. (I'm not sure how you can get hill climbing training done if you live in Illinois.) The Pyrenees rise rather abruptly, so you generally will not have long, gentle climbs. If you go in late summer, there will probably be little to no snow left. The first time I biked there was in September, and the 2nd time June. The high mountains around Andorra were completely barren in September. The Pyrenees were full of snow in June and were prettier. I had hot, dry weather both times. The western Pyrenees get a lot more precipitation than the eastern Pyrenees, so they're greener.
Nice towns include Perpignan, St. Jean-Pied-de-Port, & St. Jean-de-Luz in France, and San Sebastian in Spain. The valley headed west from Perpignan thru the town of Prades is delightful. From the Spanish-French border around Irun/Hendaye, do NOT ride the main road into San Sebastian. Either take the pretty but very hilly coastal road, or put your bike on one of the commuter trains into San Sebastian for a mere 1 euro. The main road is awful for cyclists and quite dangerous.