I think the fastest way is to go down 395 to Lee Vining then take 120 across but don't do what I did and wait until you get to Lee Vining to find out that no only was Tioga closed, but so was Sonora and I had to detour back north Halfway to Tahoe and go ove Monitor and Ebbetts passes. +6 hours.
My cousin was coming up via bishop (She lives in Orange County) and she didn't know about Monitor and Ebbetts so she DID detour back up through Tahoe. +10 hours.
This all took Place in May of 2005.
I actually spent a week in Tahoe this May and couldn't have had better weather and there was no mud. I think you'd be fine to go in June.
The Timber Lodge is Beautiful but if you can, exchange in to the Grand Residence Club.
Finally, Most Campsites in Yosemite only have bear-boxes big enough to accomodate the average-size cooler and it's prohibited to keep anything in your car that a bear might confuse with food. So Pack light. Personally I think their bear policies are a little paranoid but I guess that's better than being Bear-annoyed.:hysterical: