Greensboro is a great place to raise a family, especially if you're family enjoys parks. The city has over 5,450 acres of parks and open space. My favorite is Country Park, which surrounds a large lake that you can take paddle boats out on. It also has a trolley that you can ride on the weekends. You can find out more about Parks and Rec here.
(http://www.greensboro-nc.gov/departments/Parks/ - broken link) There's also the Natural Science Center, which has an Omnisphere Theater and a zoo featuring a variety of animals, including tigers. Residents just approved a $20 million bond for the Science Center, which be used to expand the zoo, add a restaurant, and build an aquarium. Also, the NC Zoo isn't too far away from Greensboro in Asheboro. The regional transit provider services the Zoo if you didn't feel like driving.
I think the city also has a vibrant art and cultural scene. First Friday
downtown is a great way to experience downtown art galleries. There are also some great museums, including Weatherspoon at UNCG, the Historical Museum, the upcoming Civil Rights Museum, and the Children's Museum, which is currently building an edible school garden for kids. Elsewhere Art Collaborative is one of the newest, and is a really interesting "living museum" exhibit.
The malls are malls. There are some stores that the Triangle has that Greensboro doesn't, like Nordstrom and Saks. But most everything you'd find in the RTP area, you would be able to find at one of the Triad's malls: Alamance Crossing, Four Seasons, Friendly Center, and Hanes Mall.
There isn't a Trader Joe's yet, although there have been a lot of rumblings lately about one opening. I've been to it in other areas and I really don't see the appeal. But to each his own. I would rather visit local markets like Deep Roots.
Harris Teeter seems to have a lock on the local grocery market, although you'll also find Food Lion and Lowes in certain areas. There's a neighborhood grocery store called Bestway in the Whiskey District near UNCG and it's pretty nice since it was bought by new owners a couple of years ago.
It's hard to say where you might be able to live without other information, i.e. if you plan to rent or own, how much you'd like to spend, what kind of school you want your kids to go to, etc. If you can provide that, I'd be happy to help as much as possible.
If you have any other questions, just ask.