We are transplants to Charleston too and love it! Where you look depends on why are you coming to Charleston. Being close to downtown and the beaches were essential for us.
Neighborhoods in the order of initial preference:
Mt Pleasant- very nice houses, nice neighborhoods, tons of shopping/restaurants/the best schools/ etc. Too pricey for us and honestly I would be tired of the traffic and living close to shops I like would have been terrible for my wallet.
James Island- This is where we ended up. Affordable house, big yard, close to our favorite beach, good schools (not ranked as high as Mt P). Downsides, we looked at a lot of rough houses in bad neighborhoods. It's strange, there will be dilapidated old houses just down the street from a gorgeous Charleston style homes. Definitely look in person, bc the neighborhoods are a little strange.
West Ashley- Like PP said, we found a few really cute older homes in WA but just decided it was too far from the beach for us.
Downtown charleston- If only we had a budget of >500,000 to get a place we really liked... everything in our budget was too rough for us.
Summerville- Much better prices and value, has it's own downtown area. Good schools but totally NOT Charleston. It is way too far away from downtown and the beaches for our taste (DH works downtown Charleston).
North Charleston- We compared the crime rate for North Charleston 7th in the country in 2007, somewhere from 50-75ish now, whereas Charleston was 167 in 2011 and then did not look at any homes in No Charleston.