All the previous answers have covered almost all the places one can visit in Jaipur. One thing I'll add, for City Palace, Jaipur, there is cost involved based on how much you wish to see. For just basic museum and courtyard entry, it's 300 rs.
There is a Royal Grandeur tour and Royal Splendor tour which is priced at 1500 rs and 3000 rs respectively. Here a guide takes you inside selective private rooms of the palace. I went for the 1500 option and it was totally worth it. For stays, check agoda.com or booking.com. We booked HR Palace in Bank Park which had a good spacious option for family room for 4 of us friends. Stay was reasonably priced and room service was very good with a lot of food options at a nominal price.