Hi Catlvr! Welcome to our Forum. We're thrilled you're here and finding good quality assistance. I moved your post to a board that's more appropriate for your questions.
You have several options when you're ready to move your family. You can move to an existing house or build your own, just like when you started the game. To move your family, use a telephone (cell or house) or the computer. Click "Move" and then select the house or empty lot you want to purchase. You'll be on the map view and there will be tags that show you which houses are occupied and which are empty. It will also show you the ones you can afford to buy. When you've made your choice, click on Purchase or whatever the button is. You'll get a window that shows the members of your family with arrows by each of them to click them over to the new house. At the bottom will be an option to either sell or pack the furniture. If you sell it, you'll lose all of it and your family funds increase accordingly. If you pack it, you'll find everything (even cabinets and appliances) in your family inventory when you arrive at the new house. From that point, you can pick and choose what you want to use, store, or sell. The new house will also be completely furnished, so you'll have plenty. The game makes it very easy to move, really. And yes, your Sims will retain all of their progress in skills and career. None of that will change.
Have fun and let us know how it goes!