I'd suggest the Sierras anytime from Sept 1-25 (or so); or Zion anytime from Sept 25 to Oct 31.
In the Sierras you have almost too many choices to list. Access the area from the EAST side (Independence, Bishop, Mammoth, Lee Vining, etc.) for the most spectacular hiking. Order the "Sierra South" book for specific route ideas.
Personally, for September there is NOTHING better for newbies than a 3-4 day, take-your-time loop starting at Agnew Meadows (near the town of Mammoth) and looping to Lake Ediza, Garnet Lake, Thousand Island Lake, and back along the High Trail while the aspens are turning color. That place is very crowded in August but it clears out after Labor Day.
Every time I take a "new backpacker" to that place, I acquire a backpacker for life.
If you have more than 3-4 days then you can do a second trip. A spectacular second trip (~3 days) would be exploring the Sabrina Lakes Basin or the Palisades trail near Bishop. Both of those trips are profiled in the Sierra South book.
Order the "Hot Springs and Hot Pools of the Southwest" book. Lots of info in there about the remote hot springs very near the Highway 395 connecting Mammoth to Bishop.
Zion is definitely a friendly place for beginning backpackers. You might do a lot of climbing; but the trails are extremely well-marked.
I wouldn't be as inclined to go in May or June. The snowpack is too unpredictable. The Sierras are more interesting navigable (IMO) when you can get into the East-side trails I mentioned without worrying about snow covering up the trail. Depending on the year, it might not be melted off till late June.
And May-June in the Southwest is usually too hot for me.