Pink city -Jaipur
Jaipur is connected by most of major cities in India by Air/Rail/Road. So cheapest source will be train. Fastest will be by air.
Jaipur is always been welcoming city. You shall always find tourist from all pars of India and also lots of foreigners.
Most of places are within city limit and can be reached with public transport if you are staying near by Jaipur railway junction.
There is self driving option also available on Jaipur if you have valid driving license. It may cost you 400-500 per head excluding petrol or advance if any.
There are entry fee for major attractions prefer to buy comprehensive pass for all site which is valid for 2 days.
Cover Hawa mahal, jantar mantar, museum in first day. City palace have seperate entry fee.
There are lots of shop on Hawa mahal road just you have to bargain hard.