If you go to Death Valley from Las Vegas the shortest route is to take hwy 160 to Pahrump. Its about 60 miles. Once you leave vegas there isn't much until you get to Pahrump. Then turn onto hwy 372 which becomes 178 to Shoshone (about 90 miles from LV). Follow 178 and it will take you into Death Valley National Park. It's about another 80 miles to Furnace Creek in Death Valley with no services along the way. So its about 170 miles from Las Vegas with few services along the way. This will take you past Badwater which is the lowest point in North America. A slightly shorter route is to continue up to Bell Vista Ave. Follow the road (Ash Meadows/State line rd) unit you reach hwy 127. Go north about a mile until you reach hwy 190 west. This will take you to Furnace Creek. If you want to go to Badwater you will then head south. Even in March take plenty of water.
There isn't a good way to get to the Grand Canyon south rim. You could take Boulder Hwy into Boulder, NV. You then have the option of taking the hwy across Hoover Dam or the new bridge. Either way you'll be on hwy 93 to Kingman, AZ. You then have the option of taking the interstate (I-40) to hwy 64 which goes up to the south rim. This will be about 300 miles one way. You could take old hwy 66 out of Kingman but it will add quite a few miles to the journey but it might be a more enjoyable road. But be prepared to take lots of water.
You could head north out of Vegas and go through St. George, UT. This will connect you with the ACA Grand Canyon Connector route. Not sure of the mileage.
I'm not sure there are un-paved roads you can take from Vegas that will get you to either destination. I know of a couple to some campgrounds on the North Rim but I'm not familiar with the South Rim.