Probably the biggest news for April will be the 3 week Sunsation event. A stage will be set up in center village with live entertainment. There's also some other events going on. A few days was awesome - we had a really warm day and it felt like the first day of spring.
April can be a beautiful time - warm temps, soft spring skiing, and lots of outdoors stuff to do. I wouldn't be too concerned with getting on the hill very early (and since your kids are in lessons it may not be a possibility anyway.) There's a good chance the snow will be bulletproof till about 10am.
As far as snow goes.. it's been a thin year. However, the terrain you're interested in skiing will definitely be open. Most years, some of the bowl skiing closes in April.
Not sure how you're planning on getting around, but the Summit Stage is free and it's a short ride to Main Street, Frisco. From there you could also get to Breckenridge.
If you're going to be here the whole time, one thing to consider is buying a season pass for next year. In general, season passes go on sale in April or May. The past few years prices have been about $300 - $350 for an adult, children and teen passes are less. To make it a better deal, if you buy the pass in the spring sometimes they have included the rest of the current season with the pass. So it's possible that buying next years pass would actually be cheaper than buying lift tickets each day - even if you never came back next year to use it!