This is a tough question, Janna. It has a lot to do with personal preference and building skill. When I first started playing Sims 3, I would move to a new house because I didn't know how to build. Then as my building skill increased, I would sometimes remodel the existing house. Remodeling is a good way to learn your way around building, as well, so you'll be able to build a house from scratch someday.
When you move your Sim to another house, you'll have the option to sell or take the furniture. Again, this is personal preference. Everything from the existing house will go into your family inventory if you choose to take it. Once you're in the new house, you can use what you want from it and delete the rest for some extra cash. It's fun to live in the different houses, too. You'll probably end up doing a little of both, moving and remodeling.