"[Full Disclosure - I guided for Shasta Mountain Guides for several years]
This year is light on the snow and so a late June trip would be pushing it for avalanche gulch. YOu might be able to do it, but it depends on the weather between now and then.
Try to get up there as early in June as you can. Memorial Day weekend is a zoo. The people are the greatest danger, so avoid that weekend if possible.
As AG becomes bad (too much rockfall, for example) most people consider it an end to the climbing season there. Guide companies, however, move to the West Face and then eventually the north side in August. Technically more difficult and much less popular.
There is a lot more to mountaineering than just the technical stuff. Dealing with altitude, camping on snow, melting water, extreme variations in weather, extreme weather, monitoring speed and health, risk assessment, and so on. I'm not trying to dissuade you from attempting this. I just want to be clear. It's an amazing mountain and a great climb. Some people might argue that you won't learn until you try.
If you don't hire a company to guide you, at least consider taking a 1 or 2 day intro to mountaineering course to go over ice ax and crampon technique and snow camping basics. Guide companies, university outdoor ed programs, probably even REI offer these. And it doesn't have to be on Shasta. Learn the skills first, then come apply them on Shasta.
Only three guide companies have permits for Shasta:
Shasta Mountain Guides (SMG)
Sierra Wilderness Seminars (SWS)
Alpine Skills International (ASI)
I'd estimate that Shasta Mountain Guides handles more than 90% of the climbers, with SWS and ASI doing the remainder, occasional trips.
All other companies sub-contract through one of the above three. REI and California Alpine Guides go through SMG. As do fundraising groups like: Big City Mountaineers, Summit for Someone, Breast Cancer Foundation, American Liver Foundation, Climb to Fight Breast Cancer (Hutchinson Center), and so on.
There is a lot of good info about your climb, on line and in the town of Mount Shasta. Stop by the Fifth Season to rent gear and get up to date info. There is a ranger camp at Helen Lake (around 10,000'), where most private groups camp.
That guy that you scored a frame from & couldn't even be bothered to give a single red penny to? Ya, he had a stroke & is just now welding again... you're the epitome of hippy dirtbag consumed by selfish intent. Accountability is a *itch...
BTW, that same guy made one of the 1st winter ascents on Shasta