Hello adriane,
I've traveled some of your route and am relatively familiar with the parts I haven't driven. If you take I-80 to Salt Lake, I-215/15 south to Provo, and US /191 to I-70, you'll cross several passes, and the passes are where your principal weather situations are likely to be:
Donner Pass leaving CA- 7,056'.
Pequop Summit east of Wells, NV- just under 7,000'
Soldier Summit along US 6 between Provo and Price, UT- about 7,500'. There is a fair distance of high ground between Provo and Soldier Summit, too, as US 6 passes through the Wasatch Range.
Vail Pass in CO- 10,600'
April weather is anybody's guess at this point, but without much doubt, the above-noted places would be where snow and ice would be expected more than at other places. The whole of the trip from the western slopes of the Sierra is over 4,000' with pretty much everything from Elko nearly to Wendover over 5,000', and practically no areas from Provo to Silverthorne being less than 5,000' (Green River and Moab, however, are down in the 4,000' range).
So, those are probably your crux points to look ahead to weather forecasts for. Both UT and CO have a substantial array of live webcams to monitor highway conditions at important points such as passes. That give you the opportunity to see what the passes look like before you leave a wifi or smartphone reception area.
Safe travels,
Foy