I was in Jaipur for 4 years during my engineering days and visited Udaipur, Bikaner, Ajmer, Pushkar, Jaisalmer and even Jodhpur. So I think I am eligible to answer this query of yours.
If You love chilly nights then obviously WINTER will be the best season
If you're "neither too hot and nor too cold" kind of season loving person then SPRING /AUTUMN will be the best suitable season for touring whole Rajasthan. Good Luck! Happy Traveling!!