The Jakarta MRT is currently being constructed. At the time of this writing, only Phase I of the project has been completed (Lebak Bulus to Bundaran HI) so it probably won’t be for another few years before the MRT becomes a viable means of transport for tourists.
When we were there, we got around mostly by walking. Other than Kota Tua, all the places I visited were in the Menteng and Gambir areas. I was able to get from one place to the next on foot, though actual distances are longer than they seem on a map, especially under the unforgiving Jakarta sun.

If you don’t enjoy too much walking, then you may want to use public transportation or an app-based transportation service like Grab or Go-Jek. I used Grab Car to go from Jakarta Cathedral to Kota Tua, then Grab Bike to get back to our hotel. Grab Bike is more fun and less expensive, not to mention faster because motorbikes can weave through Jakarta’s heavy traffic.
If you want to travel in the cheapest way possible, then going by the Transjakarta Busway is probably best. I didn’t use it but the Transjakarta Busway is a network of air-conditioned buses that run on reserved busways so they aren’t slowed down by traffic. According to their website, fares cost IDR 2,000 between 5-7AM and IDR 3,500 at all other times. You’ll need to pay for fares with a reusable stored value card available for purchase at any Transjakarta station (IDR 40,000 with IDR 20,000 credit). You can check the Transjakarta website for a network map.

If you’ve never used it, then I suggest planning your route using the handy Google Maps app (iOS|Android). It’ll tell you the best ways to get from point A to point B using any form of public transportation. We use it on every trip and it’s always accurate and very reliable.