At first it is very busy coordinating movers, unpacking, setting up utilities, finding a grocery and all the little aspects of day to day life.
Then there is the settling in and exploring part. In the US, I like to start at the local library. It seems I can really get a feel for the area from the library. They always have information boards of events and happenings. Children's story times are often a good place to meet other moms. These moms can be wonderful resources. I have learned about playgroups, playgrounds, preschools, family friendly restaurants, churches, the best place for a pedicure and made some good friends this way. (Plus, I really enjoy the library. Even before having children, Dan and I would go out for lunch on Saturday and then make a stop at the library. He would get a pile of magazines and a corner seat. I would head into the stacks and peruse. Boring for some. An adventure and whole new world for me.)
The girls first priority is usually finding a playground. Also, an easy place to meet other moms and play mates for the girls. I often like to get out in the car and just drive around. It's amazing what sorts of treasures can be found when getting lost.
When we lived in China, the first few weeks were spent living in a hotel while our apartment was being prepared. Rachel and I spent part of every day walking the local streets and alleways, nosing into shops, and watching people. Or better said....being watched by people. I never thought I was that interesting to look at but to the Chinese I was quite something. One man was walking down the sidewalk backwards while staring at Rachel and I. He almost fell over a woman sitting on the curb selling something (could have been shoes, jewelry, pirate DVDs, who knows). We never went more than seven or eight blocks either way, because I was still a little wary about getting lost and not being able to communicate.