For the first week of May you'll also want to keep an eye on the weather. I remember 2 years ago we went camping the first weekend in May. It was 80 degrees on Saturday. We left Sunday morning and 2 hours after we got home, it started rain/snowing with a lot of wind.
I think I-70 is probably the easiest interstate to use to get through the city, better than I-25 going N/S. I'd just stay in one of the middle lanes (don't use the far right since there's so many people getting on/off), and just take it easy. And agreed, don't go during rush hour or try to go up the mountain on a Saturday morning, or come back on a Sunday afternoon/evening. You can also try using I-76 if you're coming from Nebraska area, but you'll end up in I-70 anyway eventually.
Visalia native here :)