Day care is the type of childcare that looks after your child while you work fulltime. Many go up to Kindergarten age and are approved by the state to provide Kindergarten schooling (so that the child will automatically progress to Grade 1 when they leave).
Pre-school is a playgroup/nursery school, providing sessions of about 2.5 hours, 3 - 5 days a week.
Kindergarten is the first year of school, and is publicly funded.
To settle quickly, you want to look out for whatever is offered in your local community for families with young children. A lot of Americans go to organised events such as Gymboree (similar to Tumbletots), or will do organised activities at their local leisure/recreation/sports centre (often the YMCA). Mother & toddler groups are harder to come by, but you might be lucky. Coffee mornings at people's houses are called playgroups, and you really need a personal invitation to get into one of these. If all else fails, there's storytime at the library.
I would caution against travelling too far from home for your regular activities. The good thing about staying in one area is that you will see the same people at different things, then you will have something in common with them.
In the spring and summer, you should look out for the sporting activities that are offered in your neighbourhood (baseball and soccer, mostly), and think about joining a pool club.