That's about 5500 miles round trip using I-80 (iffy in May weather wise). You'll average about 50 miles an hour so an 8 hour day (that's a loooonnnngggg day) gets you 400 miles down the road. That's 2 weeks on the road, stopping only for overnights and not seeing anything along the way. I would think a month is not enough time. Six weeks not much better.
Route wise, I'd do the southern route (CA99/CA58 or US395 to Barstow to pick up I-40) on the way east and I-80 west. The snow in the mountain passes in WY should be gone by time you head for home.
This would be a grind for me. I'd just fly. Flying would probably be cheaper too unless you're a big family.