I would suggest a CYV (courtyard villa) because for one, they have a privacy fenced backyard and are excellent for dog owners. In the established Villages fences are not allowed to be installed except on CYV, which come with a fence. You didn't give us a budget, so it's hard really focus on an area or a home.
Go to the link for this website, there are 1500 rentals in The Villages (TV) listed, about a 1/3 of those are CYVs, I've set the link to that page. So look for ones that will accept dogs and the length of rental they require.
Villagers Homes 4 Rent
As far as buying, generally the further south you go, with route 42 as the northern most boundary of TV, the further south the prices will go up because the homes are newer.
To give you an idea of the prices, here is a CYV for sale right now, it's a 3BR 2BA, it was probably built around 2007. It's in the Village of Tall Trees, which is very central in the established Villages. They're asking $329K.
For a dog owner, it has no carpet, granite countertops, the lanai has been enclosed and has it own separate A/C unit, plantation shutters on the windows and it has low maintenance landscaping in the front. Cost of living in TV is very low, the amenity fee is only about $155 a month and includes a lot of activities.
This location in Tall Trees you're within a couple of hundred yards of Walgreens, Bob Evans Restaurant, and Southern Trace Shopping Center and what we call, bank row. It's very convenient and you can drive a golf cart to most of these places and leave your car in the garage.
https://image.thevillages.com/api/ul...pg?w=648&h=400
https://image.thevillages.com/api/photo/800586.jpg
https://image.thevillages.com/api/photo/800578.jpg
https://image.thevillages.com/api/photo/718192.jpg