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Day 1

Take Namma Metro (Bangalore's Metro Rail) and get down at KR Market station.

[#1]Start with the Krishna Rajendra Market (KR Market) area. At one end of this circuit is the Summer Palace (Tipu Sultans Sultans summer palace) and the other end is the Bangalore Fort. The summer palace is a 2 min walk from the metro staton. Bangalore Fort is a 5min walk in the opposite direction.

[#2] The next metro station is Lalbagh, on the edges of the Lalbagh Botanical Garden. This is a 240 acre parkland with centuries old trees, Glass House, Lakes and a watchtower at a mini trek to the hilltop.

[#3] About 2km (15-20min by cab) west is Sri Gavi Gangadhareshwara Cave Temple ( Devastana ), one the cities oldest temples. 1km south is the Dodda Basavana Temple (Bull Temple).
[#auto]Take an auto-rickshaw for these short commutes.saves time.

[#4] Around the Nandi Temple is the Bugle Rock Park.

Pick and chose what interests you from the above. All this will easily take a day's time.

[#food]Puliyogare Point, Vidyarthi Bhavan , Kamat Bugle Rock etc are local restaurants in the vicinity with Bangalore's edition of south Indian vegetarian menu.

Take Metro train to Jayanagar and with to 4th block shopping complex (if a bit of shopping is in your agenda ). The popular night out food street in VV Puram. Get down at the National College metro station and walk to the food street (10min walk).

So, that's about Day 1. You were almost around the vicinity of the Metro route ( green line ) to explore this part of Bangalore.

Day 2

[#1] Start with the Ulsoor Lake and its environs. Ulsoor is between the metro stations Trinity and Halasuru. [#auto]Take an auto-rickshaw for the lakeside from the metro station (2km).

[#2] About 5km west of Ulsoor area is the 19th century Bangalore Palace.

[#ticket] Bangalore Palace is a is a privately owner property. Ticket prices are a tad on the expensive side

[#3] 1½ km south of the palace is the National Gallery of Modern Art.  This one of the three National Galleries of Modern Art  ( the other two being in Delhi and Mumbai). Also they regularly conduct special programs like contests, commemorative events etc.

2km south of the art gallery is the Cubbon Park area, with parks, technology museum, heritage buildings also so on....

[#food]Yauatcha Bengaluru, Indian Coffee House, Urbane Solace, Mangalore Pearl, The Bengaluru Brasserie, Church Street Social,Koshy's Parade Café ...all are popular restaurants in this area an few kms apart

[#4]Visit the 19th century St. Mark's Cathedral with massive Baroque architecture. Built in 1812, this  cathedral is  modelled on the lines of the St Paul's Cathedral, London, with a dome, semi-circular chancel and roman arches.

[#5] Kids will like the Visvesvaraya Industrial & Technological Museum (3hours needed). It's a hand on museum with tons of working models, the visitors can operate.  Next to it's the Venkatappa Art Gallery, where the high watermarks of the regions sculptural art on display in an equally impressive building.

[#6] Walk towards the Vidana Soudha ( the secretariat building) from the art gallery area through the leafy Cubbon park , covering the Attara Kacheri , Sheshadri Iyer Memorial Hall. This is a atmospheric place , especially in the evening.

Again, the above recommendations are a bit on the higher-side to cover in a day. Trim a few based on what interests you.

Conclude day two in the MG Road and adjacent Brigade Road area, to merge into the nightlife scene of Bangalore.

[#shopping]If you want the nightlife to be a local shopping expedition, head to Commercial Street (clothing of all genere) .

That's about the whirlwind 2 day tour of Bengaluru city.

To say it in Kannada, Enjoy maadi!