VERY easy to do this trip independently ...the best spot in recent years is Ranthambhore for the tigers. Spent four days there last winter and saw FOUR tigers and one leopard ...talking to the guys on organised trips, it seemed that winter tigers were seen on more than half of safaris into the national park (so plan two or more trips!). Incidently, one of the best spot for birding is along the entrance road, not inside the park - simply for the reason you can walk and see the birds much easier. All trip reports you might read tend to say this too. BUT, it was here that I saw one of the tigers and the leopard - very nice to be strolling along, then to see one of these great cats, but obviously there are safety issues.
It seems the cats are common in this fringe zone, but so too are quite a few people walking up and down the road - there has not been an attack here on people, and the opinion is stay on/near the road and the tigers will move off.
It was planned as a birding trip, but the tigers were absolutely the highlight number one - the tiger I saw in this fringe zone was under a tree in which I saw a White-naped Woodpecker, a real stunner too. The leopard was standing on a cliff top in the evening sun looking down at me ...and as I was on foot, I was very happy 'cos after watching it for a while, a jeep appeared on the road - I flagged them doen to show the leopard, then got rewarded with a lift back to town, saving the 10 km or so walk
If sounds a bit dodgy for you on foot, very easy to arrange jeeps etc to take you into the park.
Also went to Corbett for three days - harder to see tigers there cos vegetation is thicker. I think others saw tigers each day while I was there, but I didn't.
Not in India, but I think the BEST place to look for tigers, rhinos, birds and everything is Chitwan in Nepal - totally amazing and, with a guide, you can walk everywhere and I didn't rtent a vehicle at all here. High tiger density makes it rather exciting walking here, plus even higher population of rhinos - I have a (long) trip report, if you're interested I will sent it via e-mail