Rajasthan is one of the biggest state in india with 4 different geographical locations. From delhi you can first visit Jaipur via train and cover ranthambore and nearby areas, from Jaipur can take a train to bikaner, visit bikaner for a day and can go to Jaisalmer. Jaisalmer is good for 2 days, from jaisalmer take a train for jodhpur. Jodhpur could be covered in a day or 2 and from jodhpur you can pick a bus for Udaipur. Udaipur, chitorgarh and kumbalgarh would require atleast 3-4 days. Back to delhi. You need to check the train timing and count travel hours as most of the places would require minimum 5-6 hours.
Rajasthan is very beautiful, specially in winters when you don't have to worry about the temperature and focus only on the places to go. Some of the must visit places are Jaipur,Udaipur, MountAbu, Bikaner, Jaisalmer, Chittorgarh etc.
To stay within your budget - eat local food
stay in hostel
take public transport you can plan to visit Jaipur, Udaipur, Mt Abu, Pushkar and if you have more time, you can plan to visit Jaisalmer, Jodhpur as well
If you really want explore rajasthan, you should prepare your itinerary in such a way so that you can cover some offbeat places like shekhawati region(bikaner and around, garadia mahadev near kota) alongwith the usually visited places(jaipur, jodhpur, udaipur, pushkar, jaisalmer etc) because at touristy places, most of the stuff is for tourist and you miss the realistic side of it. you can can visit haunted bhangarh and kuldhara village to have some adrenaline rush. travel in local buses or trains, have local street food, avoid fancy places to stay and choose hostels. happy travelling
1. Take a train to Jaipur, explore for 2 days.
2. Take a train again to udaipur, explore for 2 more days.
Always prefer night journey, it saves lot of money.
If you have money, head straight to Jaisalmer, if not come back to ajmer and pushkar. They to travel most by train and book hotels on the spot.