For Shasta, the prime time is mid-May to early July. You want the snow/ice to climb on. When it melts, the cobbly talus is awful to try to walk over. I went one year in August--never again!
And by all means, you need an ice axe and crampons. You want to be climbing in the early morning while the snow surface is icy, and so crampons are just the ticket. Coming back down in the afternoon, conditions can be softer, and glissading is a nice way to go.
