Dordogne

In my limited experience I can't imagine the area without a car. Sarlat is a common base for many and if trying to save money you could limit car rentals to individual days that you plan on exploring outside of Sarlat. You may find an inexpensive place to stay in Sarlat or save money on lodgings by finding a smaller more remote village with inexpensive accommodations to offset the expense of a full time car rental. Definitely take a GPS.
It is an amazing place and well worth the effort to get there.

I'm not sure exactly what you have in mind. If you are a serious walker, there are trails throughout the area and I'll bet you can make a pretty good itinerary for yourself, or on a bicycle. But your question about a "base" makes me suppose you are thinking about day trips, not hiking point to point. It's going to be more challenging to get around this region without a car. If you're in good shape a bicycle might do the trick. You can also hire a driver or join up with a local tour. Otherwise it does not make sense to think in terms of a single base. You'd probably want to move about a few times, shifting your base from Sarlat to the Vezere valley to (perhaos) one of the small towns on the Dordogne. Personally, I'd want wheels, to be able to see all the good stuff.

Thanks for the replies. Actually I think hiking point to point would be cool. Is there any kind of public transportation such as a bus to get to some of the places? I have seen self guided walking tour packages, but I don't know the terrain. Some of the walk are 10 to 16 km. I don't know of that is feasible. It seems to me that the terrain is very hilly.

As I said, bus and train service is very very limited. This is not a region to see by public transportation unless you have a lot of time. My advice: If you can't hoof it or hire a driver, rent a car. If you want to see for yourself what is possible, use the German rail site. It includes the SNCF buses. http://www.bahn.com/i/view/GBR/en/index.shtml

Daughter and I stayed in Sarlat for 4 days without a car. Found out very quickly the lack of bus and other public transportation. We did a couple of great day trips with Allo Phillipes taxi service and did our own walking tour into the countryside on another day.
I think doing the organised walking tour package would be the way to go.

Thanks for the responses, everyone. It really is helping. I am thinking about a trip in May, so I have to decide pretty quickly.
Just one more request please? Are there certain villages that any of you would recommend as "Don't Miss"?

Besides Sarlat, my "don't miss" places would be: Beynac, hilltop castle and beautiful river setting; Castelnaud, castle ditto plus lots of signage and exhibits; Domme, hilltop town overlooking the valley; Lascaux II or one of the other caves (Font de Gaume is nearby, we also heard great things about Peche Merle to the south but didn't visit); LaRoque Gageac, one-street town against a cliff, an easy stop to or from somewhere else; Prehistoric museum at Les Eyzies de Tayac. I agree with others that driving is the way to go unless walking itself is your objective, in which case you'll have to choose routes and sites carefully and certainly won't get to all the above. If you're driving, you can save lodging costs by staying outside Sarlat.