Congratulations on the new job. There was another post recently about an Indian couple with a 4 year old son moving to the US for a job in North Natick as well. Must have a lot of people in that country
J/k...
Anyways, to answer your question - it depends on the town. Most towns require you to attend the school that your address is zoned to. If you move within the same town, you will most likely have to attend a different school too unless you move within the same school zone. If you move to another town then you will have to attend a different school district altogether.
In terms of renting in a good school district, a $1500 budget will pretty much keep you out of all the top school districts, but you might be able to find a very small 2 bedroom (under 1000 square feet) in Natick for around $1500/mo. Natick schools are generally regarded to be in the top 20% in the Boston area which is good enough for most families. If I were you, that is where I would start my search.