I highly recommend investing in a spray paint trigger such as this one before you tackle a project this large. Your fingers will thank you. I personally used this Rustolem Painter's Touch 2X Ultra cover spray paint, but whatever you choose to use, make sure that it is appropriate for plastic. I also coated mined with 2 coats of plastic primer beforehand. Overall I used 16 cans of paint for the interior and exterior of this playhouse.
Here you can see the prep process of primer, and then white paint. Make sure you allow for appropriate drying time in between coats.