Get from coulommier station to Stade yes du manor and which tickets to but

This will require taking the Transilien P line train from Coulommiers to Gare de l'Est train station, then walking to Magenta RER station and taking RER E to Haussmann Saint Lazare and walking to Gare St. Lazare train station, transferring onto a Transilien J line train in direction Earmont Eaubonne until station "Le Stade".  You can see this route on Google Maps.
 
For the ticket, you'll need a custom printed ticket (a billet Ile-de-France type paper ticket) that will get you from Coulommiers to Le Stade, Colombes station.  I'd suggest you request this particular fare from the ticket window agent at Coulommiers station and they can craft you this ticket with those two stations in mind. It may actually require two tickets to complete as this is an extremely long itinerary, taking at minimum 1h50m and may not be possible with a single ticket and must be spread over two tickets. (A single ticket is valid for a maximum of 2 hours.)  Anyways, the ticket window agent can help you figure this out.
 
I believe the price given in Google Maps (16€) is incorrect because it doesn't know about the "special allowed" transfers that can be made in public spaces for a few itineraries.  The transfer between Haussmann Saint-Lazare and Gare St. Lazare is just one of those transfers.  So the ticket price could be as low as 6€ (one-way), but it might be 12€ if you're forced to use two tickets due to length of time of travel.  (The 2nd ticket you'd use at Gare St. Lazare when boarding the Transilien J line train, if needed.)
You can buy multiple tickets for this journey in one transaction, say if you wanted return tickets, buy a 2nd ticket and use it for the return journey.  (Or multiple tickets for use over multipel days.)  These tickets don't expire and don't begin their validity period until you validate them in a turnstile/faregate.