Hey Corey welcome to the forum.
Well in general, May and June will tend to have a lot of different hatches so fishing a dry fly can be a lot of fun. A lot will depend on the individual river system’s conditions due to snowmelt etc.
You’re probably better off zeroing in on a few places, so if you could provide a few more specifics about where you are and where you want to fish that would be helpful if you want more specific advice. You can also find a lot of info by googling the hatches for different rivers. Here’re somea general ones for Northern California, but some rivers may have some difference s in hatches and timing can vary from year to year and by altitude- with higher elevations (with colder water) often a week or two behind lower elevations.
California Hatch Chart - Fly Fishing Community
If you're intersted in fishing the Sierra's, Steven Ojai's website has a ton of info:
Local fly shops are also a great source of info, so you may want to seek some out.
The folks at Leland in SF are pretty wired into the fishing in Northern California
About Leland Fly Fishing Outfitters
As well as they guys at Kienes in Sacramento Kiene's Fly Shop - Sacramento's Largest Fly Shop - Fly Fishing @ It's Best!
You might also consider picking up a a book like this one:
Amazon.com: Northern California River Maps & Fishing Guide: Staff, Ray Rychnovsky, Esther Poleo: Books
If you’re new to fly fishing you might also consider koining a local chapter of Trout Unlimited or a club affiliated with the Federation of Fly Fishers
You can search here for FFF clubs to see if there’s one near you:
Locate a Club
and here for TU Chapters:
Council/Chapter Search | Trout Unlimited - Conserving coldwater fisheries
Joining a club or TU Chapter is a great way to get up to speed. They’ll habe casting clinics, informative meetings, and group trips to local waters.
Keep asking questions here, I'm sure folks will jump in with more info.