The best way to visit Ranthambore from Mumbai is to take the train. The train station is a mere 20 mins away from the park whereas if you fly into Jaipur, it's a 3-4 hour ride to get you to Ranthambore national park. No guesses which is the better option. There are frequent trains from Mumbai to Sawai Madhopar (Ranthambore station) and a 12-hour overnight journey gets you to Ranthambore the next morning.
It is advisable to book your safaris in advance. Since ours was a last minute trip we could not do so and ended up shelling a lot more for our safaris. Here are some tips -
Safari booking at Ranthambhore National Park is controlled by the Dept. of Forest (Govt. of India). As per the current policy 20 jeeps(06 seater) and 20 canters(20 seater) are allowed to enter the park per drive (morning and afternoon). We booked through our hotel and paid around INR 1200 per person for a canter and INR 2800 per person for a jeep. Keep in mind the rates vary and if you book directly through an agent you would probably get it cheaper, as we soon found out the next day.Of course the best option would be to book directly online on the Rajasthan govt's wildlife portal http://www.rajasthanwildlife.in/