One day staying at mammoth lakes in June . how could I spend my day. Thinking of mountain biking down the mountaining.

If the Reds Meadow road is open I would take the shuttle down to the trail head for Rainbow Falls. You can also continue hiking down to the lower falls but it's not as spectacualr.
Or stop at the trailhead for Devils Postpile, visit that, back track to the San Joaquin River, cross it at the bridge and turn left toward Minaret Falls.
We've done the biking down the mountian. Fun. We needed lots of bandaids. Ok it was just me who needed ltos of bandaids.
You could ski if it's still open for skiing, which it usually is (and mountain biking isn't open yet). Go to www.visitmammoth.com to see the wide range of activities in the area.
There is an earthquake fault right near mammoth mtn which is worth a short 30 minute stop.
Depends on how much snow we continue to get, but I'd go for a hike, then somewhere for a massage, then Whiskey Creek's amazing half off happy hour (everything on the menu) from 4:30-5:30pm.