First off, let me say: I come from a Professor Layton and Ace Attorney background. I live and breathe puzzles, but that's not to say that I'll take whatever I can get. My feelings towards the core factor of the game, puzzles, in Puzzle Agent are a bit mixed, but lean way more towards the positive end than the negative.
Let's start with the negative. First off, some puzzle instructions are a bit difficult to understand or are surprisingly vague. However, this only happened to me a few times, and was far from the majority. Secondly, there were some jigsaw puzzles that, if you've played the Professor Layton games, are like the ones where you have to fit items into an area without them overlapping. There are a few here too, but it's odd in pieces automatically click together when placed correctly, and they STAY together, which greatly lessens the challenge. Those puzzles could definitely benefit from a better system, especially one where the general shape of objects were outlined and could snap together even if they weren't placed correctly.
Other than that, I found the puzzles enjoyable. A few were quite easy, yes, but others took me a few tries and for some, I had to pull out a good ol' fashioned piece of paper and pen to plot it out Layton-style. Frankly, I don't understand how some people claim to have spent so many hints trying to solve them; I ended up using only 5 or 6 throughout the entire thing. There are 35 puzzles in all - I'll get to that in a moment - with 3 hints per puzzle, so with well over 100 hints you can imagine that 5 is far from a lot. To each their own, I suppose.
I've also seen people claim that the puzzles are placed erratically and interrupt the story, but honestly? I've felt that numerable times playing Professor Layton. That is the consequence of puzzle games. They WILL interrupt the story if you choose to do the non-plot-important puzzles, and even some of the necessary ones. That's just how it is. However, especially during the climax of the game, the puzzles begin to weave into the story a bit more cleanly than before, and in my experience, are appropriately some of the easier ones, so you can get through to the ending quicker without having to spend half an hour on one puzzle.
I think the game's biggest downfall for me, personally, is LENGTH. It is simply too short. Like I said before, there are only 35 puzzles. The average Professor Layton has over 100. It only took me a few hours to get through the entire game, and it leaves you on a VERY unsatisfying cliffhanger ending. Granted, it does pick up again in the sequel, but it literally goes right from the climax to ending in the first, with practically no conclusion. It was surprising, to say the least. Once the credits popped up I said "that's it?" out loud. I was genuinely surprised.
It's a bit cheap to me to separate these two games when it certainly feels like it should be only one, twice-as-long game, but I have no played the second as of yet, so I have no info on how the story of that unfolds or if it is completely related. At the very least, there should be a package including both games so it can be one long, consecutive playthrough without people having to wait until they get money (i.e., me) to finish a story that should have been wrapped up already.
The art style, characters, storyline (it's one of the "out there" kind of stories, but it's strangely fun at the same time despite its absurdity), and audio are superb. Music matches the in-game events wonderfully, but don't distract during a puzzle, as it should. I enjoyed the protagonist and the supporting cast, and their voice acting was wonderful as well.
I fully recommend this game to new players of puzzle games, or veterans of the Professor Layton series. It may be a bit clunkier than the latter in some respects, but it is definitely worth a good playthrough. You're bound to get some enjoyment out of it.
EDIT: Oops, I believe there are actually about 37 puzzles in the game; I had missed two during my first playthrough. Also, I'm going to add that I am currently on my third playthrough of the game, and I still enjoy it. I'm going to add that the game is somewhat fast-paced, or at least the cutscenes are, so even though puzzles will be extraordinarily easy to solve, I'd recommend multiple plays because I've noticed several things that I didn't recognize the first time around, buuut perhaps that was because I was intentionally reading into things because I was analyzing Nelson Tether's character a bit. In any case, definitely worth another look at if you enjoy playing it.
One final point I'd like to bring up is that of the somewhat "rating" system assigned to the puzzles. Depending on how many wrong answers you give (resulting in the anger of millions of taxpayers) and how many hints you use, you get assigned a rating of sorts. Perhaps unintentionally, but hilariously, there is some pressure on the player to answer correctly on the first go, so you can expect to unconsciously triple-check your answers before you submit. It's really not something to fret about - at least to me, it's more humorous than anything else.