I would first recommend talking with the school's Principal. Hopefully she'll be 100% supportive...
Next would be to approach at least a couple of people and see if they would be willing. Once you have a team then you can start moving forward.
Go to your town hall and see if they have any information from the previous group on file. If not you might be able to approach another group in the town, or just use what is on the site here.
You can form prior to obtaining your non-profit status, but you'll want to file that right away.
IMO, for the first year, your focus should be on raising funds, creating a set of Bylaws, obtaining your non-profit status (501(C)3), and talking with the parent/staff and Principal about what can be improved. I would recommend against tackling any major project, such as a new playground, in your first year. There's just so many other areas that need to be focused on first. Next would be membership. You'll want to start promoting the group and communication is the way. A website, newsletter, flyers about the new group, etc., should all be used to get the word out.
Then, once you have all this done this year you might want to end the year on a high note. Possibly take on a big event, such as Field Day. This might be your first event, but it will show the parents that the group will make a difference. Then come next year, with funds in place, a basic budget, priorities for the year, elections having been properly held, etc.; you can start the new year off with a bang, maybe a back to school event.
Good luck,
PresidentJim