I am looking for a place in the Caribbean that 1) has nice villas or multi-person accommodations that can be rented, say, from a Thursday to Tuesday and 2) lots of nice things to do "outside" of the resort, i.e, someplace with nice tourist sites, good restaurants, a nice downtown area and 3) is not prohibitively expensive either to fly or stay there. FYI: We tentatively have a party of six and are planning to go in late January. Any suggestions welcome.

Your requirements can be met in many of the islands of the Caribbean, but you might be interested in Barbados, St Lucia, St Maarten or Aruba for starters. January is high season so everything and everywhere will be more expensive.
You might find some more useful info here

http://www.destinationcaribbean.net/...amp;Itemid2=70

and be able to narrow your search a bit.

you could look in the playa del carman area, approx 90mins south of cancun, which is easy to fly to from most places. look at locogringo.com for loads of info. this area has tons of activities (golf/tennis, snorkel/diving, caves/cenotes, zipline, fishing, kayaks and sailing, tours of ruins and 2 eco marine parks - xcaret & xel-ha).
pdc has restaurants, bars, nightlife and shopping.
here's a sample house just south of pdc: http://www.locogringo.com/playa/casa...=Casa%20Grande
click on the little airplane at the bottom of the villa description - shows you where villa is.
you can email locogringo, tell them what sort of house you need, the locations acceptable and they will send you a list of properties available for your dates.

st. martin might be another option. easy to fly to from most places, large town to play in. haven't been yet so can't tell you much more about it.

or on st. thomas you could look at www.secretharbourvi.com a condo style resort on a great beach, with snorkeling and a dive company. charlotte amalie is a good sized town with tons of shopping. plenty of great restaurants. charlotte amalie is about 45 mins from secret harbour.

stay in the Soufriere region in St Lucia to experience waterfalls, volcanos, hiking, mountains, rainforests, wind sailing, spectacular snorkeling

There was a recent post here about great resorts for $150/or less per night. I tried to pull it up, but can't find it right now. Anyway, they listed a property in Grand Cayman that sounds like it would be ideal for you. I was in touch with them, and the property offers several standard to deluxe apartments on the water, although it is not on Seven Mile Beach. I'm so sorry that I can't find the name. Thought I saved it to a file.
We have been to Grand Cayman several times, and I really think it offers all that you are looking for and more, although restaurants can be quite expensive. Maybe someone else here can pull up the post? I will look for it again later today.

Actually found it! It's called TURTLE NEST INN AND CONDOS on Grand Cayman. Good luck!