If you pick your hotel wisely, you can get around to most of the sights of Solo on foot. It's a very walk-able city, with wide sidewalks and the main streets have side roadways that provide additional space for walking.
The traditional pedicabs, called becak, still ply the streets of Solo. You'll need to negotiate hard to get a good price, as they will start with a very high fare when they see western tourists.
Taxis are a fast and cheap option for going any distance. You may be hard pressed to flag one down, but they are usually queued up outside hotels and shopping centers. You can even negotiate with some taxi drivers to take you around for a day of sightseeing.