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The public transport in Jaipur is nothing to write home about. Althought the city metro line is due, it runs only on a limited route as of now. for many visitors, the tuk-tuk (or "auto-rickshaw" as they are called locally) is perhaps the best means for local travel. But one need to bargain a bit, and depending the time of the day, the fare quoted may be a bit arbitrary. For local travel, sometimes may be cheaper than hiring a tuk-tuk, taking cab from a taxi hiring service such as Ola (and also perhaps Uber) may be really convenient.
The Rajasthan Tourism Development corporatation (the state government tourism department) cab hiring services for local tour may work out more expensive. If you are staying away from the Pink city area, then perhaps you could save on hotel rates.
One could cover most of the city landmarks in two days - the first day to vist the palace complex inside the Pink city, Jantar Manar (just next to the palace), Hawa mahal, and Albert Museum (a short tuk-tuk ride away), and the second day to visit the neighbouring forts - Jaigarh, Nahargarh (they are nearby, and Amer Palace. One could explore the market in the Pink City (expecially Johuri Bazar) on the first everning, and may be make purchases on the second evening.