Take one of those city bus trips from Mysore city (Bus No 303,303A). They operate quite frequently. The evening trips could get a bit over crowded. The bus drops you right next to the garden entry ticket counter.
Mysore city to Brindavan Gardens distance is about 18km. If you are driving, take the KRS Road out of the city. At It's a 30min drive.
After about 10km from Mysore City center, you'll reach a T junction called Pump-house Junction. Turn left for Brindavan Gardens. The right turn will take you to Ranganathittu Bird Sanctuary, Srirangapatna and the Bangalore Mysore highway.
After turning left, you'll cross Belagola village (take diversion here for Balmuri and Edmuri Falls) and hit a Y fork. Take the right again for Brindavan Gardens. The left form will take you to Yelwal on the Hunsur Road.
The left fork will eventually end near the KRS Dam site. There is a toll fee collection booth before a long bridge. There is a massive parking lot right next to the entry to the garden.
On you way back, if you are heading towards Bangalore, keep driving straight at the Pump house Junction. You'll reach the Bangalore Mysore highway near Srirangapatna.
Srirangapatna diversion on SH17: Route map to Brindavan Gardens (KRS), Balmuri & Yedamuri Falls, Ranganthittu bird sanctuary and Srirangapatna from Mysore. Note everything is shown along the Cauvery River. Srirangapatna is practically an island formed by two branches of Cauvery River. A number of tourist attractions are located in Srirangapatna. If Srirangapatna is part of your Mysore itinerary, keep it as the first or last item for obvious logistics advantage. Keep in mind, you may need a couple of hours minimum for covering the popular attractions of Srirangapatna.