Most "real" camel safaris are done from Jaisalmer, which is hell and gone from just about anywhere in Rajasthan but still a must-visit. There are also apparently 2 or 3-day camel safaris from Bikaner but I don't know if you're going that route. I think there are camel rides available in Pushkar but the Thar Desert (Jaisalmer) is supposed to be better. I didn't have time to do that on my last trip there.
From Mumbai I would go directly to Udaipur to feel as though you've been plopped into the middle of a fairytale. From there depending on how much time you have you could do something like this (all easy on trains or buses):
Udaipur to Jodhpur (the Blue City, try a makhania [sp varies] lassi while you're there-- sooooo yummy, and Mehrangarh Fort is spectacular)
Jodhpur to Jaisalmer (the Golden City, so named because it's in the middle of the desert... they have a spectacular fort too)
Jaisalmer to Ajmer
Ajmer to Pushkar (beware the fake sadhus and self-righteous "India is so, like, spiritual, man" hippies)
Pushkar to Jaipur (the Pink City... and Amber Fort is, well, spectacular too)
Jaipur to Agra
*side trip if you have time: Agra to Varanasi, potentially the oldest continually habited city on earth (and it looks it), and the holiest hindu city*
Agra to Delhi
... that's the route that would make the most sense (to me anyway) with little backtracking, but it's entirely up to you.