The sign of a well made competent musical is that it can charm an audience who isn't particularly excited about the music into having a good time, getting them interested in the characters and the overall emotion and energy of the movie. "Mamma Mia!" isn't one of those films.
There are several problems with "Mamma Mia!" that make it something that will appeal to fans of the music and the stage musical, but make it a kind of throwaway experience or downright irritating for the uninitiated.
The director, who helmed the stage play, was the wrong choice for the movie. Very wrong. The actors "ham it up" and directed as if they're playing to a stage audience, entering the film all squeaking and sqwaking making sure the cheap seats in the back row are getting their money's worth. They try to hard, coming off as if they're coming out the screen and trying to bang you on the head to make you have a good time.
They didn't need to. Indeed, "Mamma Mia!" is a fluff piece, but, played a bit more subtly and straight during the sequences in between the musical numbers, it could have maintained a steady ebb and flow of energy that wouldn't have seemed so tiring and ingratiating.
The musical numbers themselves showed no structure or imagination. Mostly, the actors are static and the camera moves around them or ideas are thrown about and the numbers just don't really build to a climax with the song. For a musical, it's the most scattershot directing one can imagine.
So, here's a couple of examples of how I would have fixed the musical numbers.
In "The Winner Takes It All", Streep's character has a frank talk (or sing, as it were) with her former lover Pierce Brosnan on a cliff at the bottom of a larger hill where a crowd is headed up to her daughter's wedding. Brosnan doesn't have anything to do in the scene except just stand there and it's on Streep's shoulders alone to carry the song. Then, when the song ends, she has to run up a bunch of stairs to her daughter's wedding. Very dumb staging.
It would have made more sense to have Brosnan call her aside outside the church and do the song there. The scene would have turned into more of an interplay between the two characters visually as they moved around the outside of the church in a kind of lover's pursuit. In the background, you would have seen, through the windows of the church or around the periphery of the building, the preparations for the wedding, making the song have more emotional impact. As it stands, the thing is just static.
In several sequences, local Greeks are used as a backup singers in the numbers. The problem is that they just look like local extras doing kareokee and just don't have anything to do except provide a visual for some backing vocals. I would have turned them into, literally, a Greek chorus that would do simple dancing or other business around the characters, then go back to being local color in the background in between the numbers. This is really apparent in "Does Your Mother Know" - the guys on the beach look like they're just milling around without any purpose except to smile and gawk at the camera.
I won't even go into the structure of the book, a problem inherited from the stage show. Basically, you've got a second act that goes on for a good 80% of the running time of the film and a third act that's about five minutes long. It makes the main part of the film excruciatingly long if you're not into the music as you wait for the thing to move and get on with things; then, when all the storylines are wrapped up in a couple of short musical numbers, it feels like the authors were tired of riffing and just wanted to end the thing - it's a story arc as much as a straight line that just goes off a cliff at the end.