The Eastern Sierra is my favorite part of the state.
Since it's still early spring it's possible to hit bad weather north of Bishop all the way to north end of the state. As of this post its snowing in Mammoth.
Check the 10 day forcast before you leave and even if you have four wheel drive carry chains just in case.
Now for some things to see:
In Lone Pine you have the film musuem and the Alabama Hills just behind the town. I recommend staying at the Boulder Creek RV park just south of town. One of the best places to stay with sit down views of Mount Whitney, great amenities and the staff is A1. You could stay for three or fours days and use this as you're base of operations to tour Owens Valley and go into the west side of Death Valley. Up the road is the Japanese internment camp of Manzanar with a great musuem open in the old gym building. They have some of the original barrack buildings open and a self guided tour
In Bishop stop at Schat's bakery for some incredible pastry, bread etc. I like to stay at Browntown RV park next to the golf course on the south end of town. Try and get a spot towards the back on the golf course side.
As you go north you have many great lakes and mountains to see. Mammoth will be full of snow so many of the things to see are still closed like Devil's Postpile and Rainbow falls.
North of Mammoth you could still drive the June Lake Loop and stop and see Mono Lake and their great visitor's center. Just off 395 at the south June Lake loop road is the Pinecliff RV resort. Remember, it's still winter so most if not all county and state campgrounds are still closed.
Going north towards Bridgeport not much open until you get to Lake Tahoe and Carson City.
Be safe and enjoy,
in So Cal