Can sims move to a new neighbourhood?

You cannot move sims between neighborhoods, because in Sims3 (unlike in the Sims 2) the Neighborhood is the save file*.

What you can do, is upload the family in question to the House bins, move to the new neighborhood, and then place them from the housing bin back in the world. However, this eliminates all non-household relations for the sims in question.

*Though I suppose the whole point of the World Adventures expansion is moving to foreign neighborhoods.

Save your sim to the library so you get to save your money and progress such as skills. Or save a family and home all together to the library then go into neighborhood you want and place them where you want.

I've done it enough to know it works :D