hubbard53, are you looking to rent or to buy right away? I would suggest renting first for at least 6 months, so you can get a feel for the area.
We're renting at Greystar Daniel Island -
Charleston Apartments | Daniel Island Village Apartments It's a great pet-friendly complex with underground garages, a gym, clubhouse for events, and biking/walking trails right outside the door. Very well run complex. Nice apartments. Next door is Talison Row, another upscale complex. We saw both, but Talison Row doesn't have the ponds and green space Greystar does, the apartments are more jammed together.
Right across the street from the rental office at Greystar is one of the walking/biking trails. Take some time to walk it a bit. It will give you a feel for the trail system (paved). There are trails all over the island that interconnect the neighborhoods.
From the apartment complexes you can drive down Seven Farms Dr (the main drag on Daniel Island), and check out the shopping area with the grocery store (Publix), banks, wine store, pet store, UPS store, etc. Across the street is the Family Circle Tennis Center, where the Family Circle Cup is held (women's tennis) each year, and concerts and events throughout the year.
Next to the Tennis Center is a small dog park, and a recreation area (outdoor). There's also a nice park across from Bishop England High School as you drive down Seven Farms with a volleyball court and lots of green space.
Off of Seven Farms Dr., before you get to the downtown area, take a right onto Daniel Island Drive, and drive down to the library (on the right) and further down to Smythe Park (on the left).
Further down Seven Farms (just before all the shops/restaurants), take a right onto River Landing Drive. There are numerous condo complexes that sometimes have rentals, and a new apartment complex - Simmons Place - going in there as well. More of an "urban" feel than Greystar/Talison Row. Drive to the end of River Landing Dr. (beautiful road at the end with gorgeous live oaks) and park at Children's Park. There is a trail that runs along the water on the left and right. One of our favorite restaurants is also here - Sermet's. The Daniel Island real estate office is also down there at the end and you can pop in.
As you drive down Seven Farms past the shops, restaurants, businesses, Family Circle will be on your right. Continue under the overpass, and at the light take a right onto Island Park Drive. This is the "ritzy" section of Daniel Island, but worth a drive through. It's beautiful.
If you are thinking of buying, you might also want to see one of the pools - maybe Pierce Park pool. When you stop in at the RE office, they can give you a map of Daniel Island.
There is a hotel on Daniel Island -
Charleston SC Hotels - Hampton Inn Daniel Island - nothing fancy, it's a Hampton Inn - but adequate, and a great base for exploring Daniel Island and Mount Pleasant, especially since you are short on time.
This site gives you more info on Daniel Island and you may be able to download the map ahead of time -
Island Living - Daniel Island, SC
Daniel Island is only 10-15 min from Mt Pleasant, so it will be easy to explore from there if you stay there. Mount Pleasant has the "old" section - older homes, more of a "beachy, community feel", (South of Isle of Palms Connector), and "north Mount Pleasant", which has more of a "new subdivision feel". There's no downtown to Mt. Pleasant, Rte 17 cuts through the center.
Not sure of your budget, or if you are looking to rent which I can't help with in MP, but I would check out I'on, Snee Farm, Seaside Farms, Belle Hall, Longpoint (South), Brickyard and Hamlin Plantation (mid) and Carolina Park, Charleston National, Rivertowne (CC and On the Wando - 2 different parts) Park West and Dunes West (you will need a pass from a realtor to get beyond the gates) - (North).
These should at least give you a good "feel" for the different types of developments, locations, and housing styles available.
While in Mount Pleasant, go to Patriot's Point, get out and walk the Pier at Waterfront Park. Drive through old Village and to the end of Pitt St., and get out and walk the fishing pier. Have lunch at Shem Creek - Water's Edge probably has the best food - view is good at any - Waters Edge, Red's Vickery's, and Tavern and Table.
Drive to Sullivan's Island (you could also have lunch here - Poe's Tavern is great) and Isle of Palms. Part of Isle of Palms is private (Wild Dunes) and you won't be able to get in.
If you'd like to stay in Mount Pleasant, and something a little nicer than a Hampton Inn, Old Village Post House is very reasonable, lovely (5 rooms, I think) and has a great restaurant and courtyard. It's in Old Village which affords great walking at night. Beautiful part of Mount Pleasant, but mucho bucks.
There are 2 hospitals in Mount Pleasant - E. Cooper in the South and Mt. Pleasant Hospital in the North. Numerous shops/restaurants. You should also see Towne Centre, which is very near Brickyard. An outdoor "mall" - very nicely done. While at Belle Hall, go across the street and see the "Shoppes at Belle Hall", a big shopping/restaurant complex. While at Seaside Farms, you can drive into the "Shoppes at Seaside Farms" - another shopping complex. Park West also has a small shopping/restaurant complex at the entrance, and a recreation center.
If you have time to venture into downtown Charleston, rather than drive what I would do is drive to the Charleston Harbor Marina in Mount Pleasant (at the end of Patriot's Point Rd) and take the water taxi - $10 for an all-day pass. It stops at Patriot's Point in MP, the Aquarium in Charleston, and Waterfront Park in downtown Charleston. Very relaxing way to get downtown, cheap, and great views. It only takes about 10 min if that from MP. You can get out at Waterfront Park and explore the historic area of downtown on foot very easily.
That should be enough to keep you busy for 2 days. If you need more info. of have questions, feel free to PM me.