Hope you have a good move.
We've been car free since before our children were born. Our daughter is 7 and our son is 4. We take the bus and walk most places. But since moving near the bike path we've been riding more. Every now and then we take a taxi . We are also part of a car share and sometimes rent a car.
My best advice is wear appropriate clothing and have fun. I've found car free kids in a city to be a pleasant experience.