Mysore is a great weekend out from Bangalore, with good train and bus service, or, you can rent a car-with-driver for about $45 per day including fuel. Mysore is definitely worth a visit, to the Palace, Tipu Sultan’s summer home, and the Chamundi Hills (the Brindavan Gardens are a bit of a tourist trap and can be skipped).
If you don’t mind a longer drive and would like some cool weather, you can head to Coorg or Ooty (Ootacamund), or for one of the most spectacular places in the world, you can catch an overnight train (via Hospet) to Hampi on a Friday night.
There are a number of scenic hills for trekking in about a 90 minute drive from Bangalore, such as Nandi Hills, Savandurga, or caves at Anthargange, however, if you have a full day and don’t mind an early start, try for Mysore and/or Talakad. These are both about a 3 hour drive, so if you start around 6am you’ll reach in good time. Talakad is also unusual and scenic, and can be combined with Mysore if you move fast. You’ll also get to see some beautiful countryside (and more than a few speed bumps).
Of course, Bangalore is full of lots of fun things to do, from the Lal Bagh to the Bangalore Palace. Whatever you do, I’m sure you’ll have a great time.