The ideal way to see Rajasthan is first to fly to Jaipur, Jodhpur, or Udaipur and then tackle nearby towns by train, bus, or private car. Ideally you should start at Jaipur, make your way around by car, and end up in Udaipur, from where you can fly to Mumbai or Delhi. Overnight train journeys between major cities in air-conditioned sleeper coaches are a cheap way to travel without wasting too much time or money on hotel stays. For instance, you could also start in Jaipur and work your way to Udaipur, then to Jodhpur, Jaisalmer, and back to Delhi via Jodhpur.
There are plenty of sights and day trips to be taken from each of these major destinations. Don't miss the Jain temples in Ranakpur and the Kumbalgarh fort, both of which are fantastic day-trip options from Udaipur. Two days in Jodhpur is plenty—reserve a day for seeing the city and the fort and the next for a village safari, but make sure you give yourself at least two nights in Jaisalmer—spend one night in town/fort and the second night on an overnight desert safari on camelback.
If you only have a few days to visit Rajasthan, we suggest picking one city to visit, and try to make a few day trips from there, rather than cram in several of the larger cities. While Jaipur is the most popular destination in Rajasthan for its proximity to Delhi and Agra, Udaipur is a fairy-tale sort of place, with its beautiful lakeside palaces and havelis, and the many nearby attractions.