Krishnarajapuram [KJM] (or 'KR Puram' for short) is perhaps the best connected railway station in Bangalore after Krantivira Sangolli Rayanna [SBC] (also known as Bangalore Junction or colloquially as Majestic). Here are some of your connectivity options:
1. If you exit the station on the north i.e. Platform 1, you can catch 300 series buses headed towards Kempe Gowda Bus Station (Majestic) or Shivaji Nagar in front of the ITI campus at the U-turn under the cable-stayed bridge. These buses will pass through Rammurthy Nagar, Banaswadi, Maruthi Sevanagar and Frazer Town.
2. If...
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you exit the station on the south i.e. Platform 4, you have two options. Immediately to the left of the exit is a bus stop. Here you will find 500 series buses (normal and air-conditioned Volvo) headed to Silk Board (frequent) and Banashankari (infrequent) that ply along the Outer Ring Road. These buses will pass through Mahadevapura, Doddanekkundi, Karthik Nagar, Marathahalli, Bellandur, Agara, HSR Layout, Silk Board and BTM (not all stops listed). At this stop, you will also find 300 series buses that ply to Whitefield. The buses to Whitefield will pass through Hoodi and ITPL.
3. If you exit the station on the south i.e. Platform 4 and cross the road, you will find another bus stop. Here you will find 500 series buses (normal and air-conditioned Volvo) headed to Hebbal. Buses to Hebbal stop at Tin Factory, Rammurthy Nagar Bridge, Banaswadi, Babusab Palya, Kalyan Nagar, Hennur Road and Nagavara. At this stop, you will also find 300 series buses coming from Whitefield that are headed to Majestic and KR Market. The 300 series buses are however infrequent. If you don't get a 300 series bus for a long time, hop onboard a 500 series bus and get down at the next stop, Tin Factory. At Tin Factory, you have a better chance of finding a 300 series bus headed towards Majestic or KR Market.
4. At the bus stop across the road from the exit on Platform 4, you will also get Metro Feeder buses. You can board these buses to get down at the first two stops on the Purple Line: Baiyappanahalli or Swamy Vivekananda Road. The connecting stops for these stations can also be reached through 300 series buses headed to Majestic or KR Market. Ask for "Srinivasapura" or simply "Baiyappanahalli Metro Station" for Baiyappanahalli, and "Isolation Hospital" or simply "Swami Vivekananda(SV) Road Metro Station" for Swamy Vivekananda Road Metro Station. (Hopeful note: Once Phase II of the Purple Line is completed, you can get on the Metro right across the Platform 4 exit!)
5. At the bus stop across the road from the exit on Platform 4, you will also get Vayu Vajra Volvo buses to the Airport. The airport buses are however infrequent. If you don't get an airport bus for a long time, hop onboard a 500 series bus and get down at Hebbal. At Hebbal, cross the intersection and walk to the northern end of the flyover on Airport (Bellary) road. Here, you have a better chance of getting an airport bus.

External Links:
click here Bangalore Metropolitan Transport Corporation. To use the Trip Planner, enter 'KR Puram Railway Station' for bus stop on south side (Platform 4 exit) and 'ITI Circle KR Puram' for bus stop on north side (Platform 1 exit).
click here Namma Metro - Bangalore Metro Rail Corporation Limited