My first time gluing a puzzle was with the Times Square one. I used Masterpieces glue, which comes with a plastic card-like applicator instead of a brush, which was nice. I may have used too much as the next morning I found that it had curled up at the edges. When I was applying it, I thought I hadn't applied enough around the edges originally, so I added more. In hindsight I guess I should have done 2 thinner coats. I've ordered SunsOut glue since everyone raves about how nice it is and how it goes on white and dries clear. It's been sitting with some heavy books on top of it and it's finally almost totally flat.
Monsoon Madness was interesting to do because of the irregularly shaped pieces. This one has contact paper on the back right now until I decide if I want to glue it.