I've enjoyed a few springtime rides in southern Arizona, but let me ask a couple questions (and give you a big caveat) before I get to my suggestions. First, what's a "good" daily mileage for you? If you start with the idea of 500 miles and plan to split it over three weeks, that's 25 miles a day. It's going to be hard to get from point A to point B on that schedule. Second, will you have the capability to carry water and food for a full day's ride? Also, how much interstate riding can you tolerate? (IME, 10 miles is novel and OK, and then it gets long and noisy.)
Caveat: most of my ideas that don't involve riding the cycle paths around Tucson involve 50-75 mile days.
OK, first idea. Try a couple of loops around Tucson. Tucson to Picacho to Chandler to Florence to Oracle Junction to the airport (5 days). Tucson - Green Valley - Sonoita - Sierra Vista - Bisbee - Tombstone - Benson - Tucson (5-7 days).
Second, Tucson to El Paso, or the other way around. I'd look at routing through Sonoita, Sierra Vista, Douglas, AZ, and then Lordsburg and Las Cruces, NM. You'll be looking at a couple 70-80 mile days, and the rest would be around 50 miles a day. Could be tight housing a couple nights (Sonoita and Rodeo, one place to stay in each town).
Either way, I'd expect morning lows around 40F (but be prepared for frost), and highs 60-80F. Passes will be cooler, of course. I wouldn't expect much rain, although thunderstorms can be spectacular.