There are quite a few routes to get to Karjat, its pretty close to Lonavala and Matheran, so the train is a viable option with frequent trains from Victoria Terminus in Bombay as well as Pune.
You may like to take a look at the train timings from CST (VT) to Karjat, although it may not be completely accurate. The train journey can take you upto two hours and is quite scenic after Kalyan!
Travel to Karjat by Road
If you are getting to Karjat from Mumbai by road, you would need to get to Vashi, then Panvel if you get on the Mumbai Pune Express way then you need to take the first exit on the left its the Shedung Khopoli Exit.
The problem with coming by Car is you end up paying three tolls, amounting to Rs.160 for a journey to and fro from Mumbai, but if you decide to bike it, its a fun ride without any toll and you can reach your destination much faster as you avoid the jams at the Toll plazas.
The Road journey takes around two and a half hours including some amount of traffic.
The Map to Karjat
Karjat is located at 18.91° N 73.33° and is denoted by the blue balloon below,