The old city of Solo, formerly Surakarta, is a lot less visited than Yogyakarta. Most people only come on a day trip from the other city, but there is in fact rather a lot to see in and around the city, which makes it worth spending more than a few hours in.
The city was once the center of power for some of the great Javanese kingdoms. It boasts two palaces, and scattered around the countryside are the ruins of several interesting Hindu temples.
Like most of Southeast Asia, Central Java is subject to the annual monsoons. The dry months are July to September. This is also when the nights are cooler, so it's generally the best time to go.
Solo does have an airport with limited domestic and international service. It can be reached from Jakarta, Singapore or Kuala Lumpur. If visiting Solo in conjunction with a trip to Yogyakarta, you can easily get a car hire to take you from one city to the other, or you can take the train. The Prambanan Express which makes the trip in around an hour.
You can see the sights of the city in one full day. You'll need another full day to take in the ancient temples around Mount Lawu to the east, making the ideal stay three nights. If you want to poke around the antiques market and the batik village, then tack on another day and night.
Solo doesn't have the range of accommodations that Yogya does. The best the city has to offer is probably the Novotel, which is also in the best location, right in the middle of town. You could also be quite comfortable in the Ibis right next door. Although only a three-star hotel, the Ibis is newer, and I found the standard room, while small, were very efficiently laid out to be very comfortable for a short stay.
If you stay in the center of town, then most of the sights of the city will be within easy walking distance. 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.
The major sights of the city are the two sultan's palaces. The Mangkunegaran Palace is the smaller and newer of the two, and where most tourists are taken for a guided tour. The older, larger Kasunanan Palace houses a museum within its labyrinthine sequence of courtyards, even though it is, like the other palace, still a royal residence.
Solo is also a very good shopping destination. Near the Mangkunegaran Palace is the antiques market , where you can find old batik stamps and other collectables. There's also an area of town known as Kampung Batik , where you'll find countless small shops, boutiques and factories for the quintessential Indonesian craft. Solo is, in fact, a better place to shop for batik than Yogya.
Perhaps the biggest attraction to Solo are the ancient Hindu temples in the mountains outside of the city. The most well known is probably the curious Candi Sukuh , with its Mayan appearance and racy relief carvings. Not far from this temple is the more traditionally styled Candi Ceto . On the same mountain is a rather spectacular waterfall called Grojogan Sewu . Seeing all three of these sights is a full day trip out of Solo.