Sorry I can't be more specific about a backpacking trip, but I think the Southwest (Utah/Arizona/New Mexico or Colorado) is the way to go. I would pick Utah first, but there are some great places in the other states as well. I think the Grand Canyon would be a bit too hot, even in April whereas Utah (Zion, Bryce, Canyon Lands, Capital Reef or similar areas) is in its prime then. I would look for flights into Las Vegas (which are often fairly cheap and can be combined with a rental car easily) and then just drive from there.
I think the Sierras would still be in snow. It is a great time for a ski trip, but not for regular hiking.