The ideal way to see Rajasthan is to first fly to Jaipur, Jodhpur, or Udaipur and then tackle nearby towns by train, bus, or private car. Overnight train journeys between major towns in comfortable sleeper coaches are a cheap way to travel without wasting too much time or money on hotel stays. For instance, you could start in Jaipur and work your way to Udaipur, then to Jodhpur, Jaisalmer, and back to Delhi.
Depending on how much time you have, 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. One day and night in Jodhpur is plenty, but make sure you give yourself at least two nights in Jaisalmer—spend one night in town and the second night on an overnight desert safari on camelback. If you're an art lover, consider renting a private taxi from either Delhi or Jaipur and heading to Shekhawati—base yourself in Mandawa or Nawalgarh and then head out to explore the region's beautifully frescoed havelis (mansions with interior courtyards). Ranthambhore National Park, reachable from Jaipur, is the place to get away from the hustle and bustle of Rajasthani cities and lose yourself in nature.