Most of the sights of Solo city are within a relatively small area that can be walked. The two palaces are the main items of interest, and there are some interesting market and shopping areas in between the two. The other two museums are places you can see if you have the time and interest. The sights can be seen comfortably in a day and a half or so.
Solo is usually visited as a day trip from Yogyakarta, but these usually leave a number of interesting sights unseen. To really take in all that the city has to offer, you should plan on spending one or two nights in town. Withing the city the major sights are the two palaces, the newer, smaller Mangkunegaran Palace and the older Kasunanan Palace, which also houses a museum. Near the Mangkunegaran Palace is the antiques market, where can find old batik stamps and other collectables. Lastly, take a stroll through the relatively quiet lanes of the Batik Kampung, a section of old shops and houses near the Kasunanan Palace where batik is still made and sold.