Joe,
You definitely want to hit the California Billiard Club. It's a well thought out and run pool room. In addition to GCIII's the place has 3 snooker tables (2 of which have traditional snooker cloth and the third is covered to play golf), as well as a heated billiard table. I was up there on business recently myself and spent about 4 evenings there and every time I took a table the balls were freshly cleaned and polished.
The smaller section (towards the street) is the match/tournament area, and is preferred as this is where you will find the tighter tables.
They have monthly memberships (not required) so there is a good group of regulars there. There's action if you want it. The owner's name is Chris and I'm sure he would introduce you around.
Shoreline Billiards is up that way too. Also a good room, Bob Jewett plays out of there. I liked the CBC better though. I think Shoreline caters more to leagues than the CBC.
Both are in Mountain View, about 10-12 miles from the San Jose airport, maybe 10-15 minutes away.