I'd agree with other folks that the High Sierra in late May or early June (in a late snow year) can be pretty daunting to someone with little experience. I did the PCT in 2004 in a very low snow year, and it was a breeze in mid June. However, I did take a winter mountaineering course prior to my thru-hike, to learn about safer snow travel, ice axe use, crampons, etc.
I did the Sierra High Route starting in mid-June of a very high and late snow year, and I sure would not have wanted to be there by myself or without the experience I had gained since 2004. Dangers include steep snowy slopes (and occasionally icy) with unsfafe runouts, on which we used ice axes extensively, and also crampons on some early morning pass descents. Stream crossings can also be very gnarly, either requiring much hiking upstream to find a suitable place to cross or a tree trunk to shimmy across. Both of these dangers (slopes, streams) are potentially fatal.
As far as early season gear, on the PCT I had my ice axe, but didn't need it and sent it home from Independence. On the Sierra High Route, I had Vasque Velocity shoes, neoprene socks, ice axe (Cassin Ghost), and Camp XL490 crampons. I wouldn't use heavy boots that will just get wet and stay wet, but also wouldn't use really light trail runners - a little more support is good for all the funky conditions (talus, frozen deep suncups, etc.), kicking steps, and possible support for crampons. We could've done it without the crampons, but it would've required waiting for some conditions to soften enough to descend/ascend some passes and steep slopes.
I'd suggest either flip-flopping (though I personally wouldn't want to disrupt a thru-hike that way), or at least making sure you head into the High Sierra with a couple other early PCTers. And if you carry an ice axe, you need to LEARN how to use it ahead of time.
That said, the High Sierra with lots of snow is one of the best possible experiences EVER.