It's better and more likeable than the 2011 anime series (Persona 4: The Animation), but it's still kind of awkward and lumpy. It hasn't even considered that you might not be familiar with the franchise, for a start.
Explaining the background's going to take some time. Persona is a role-playing video game series and a spin-off from the same developer's Megami Tensei games. The first Persona game came out in 1996. Persona 4 was the fifth game in the franchise (don't ask), released in 2008/9, and it was adapted into a 26-part anime series in 2011. Persona 4 Golden is a portable version of Persona 4, developed for the PlayStation Vita. (The developers deliberately avoided the PSP after being forced to cut content from Persona 3 Portable.) Persona 4 Golden is more than just the same game on a new platform, though. It has new features and new content, enough to justify a new animated adaptation.
There isn't a word to describe Persona 4: The Golden Animation, aka. P4GA. It's not a sequel. Theoretically it interleaves with the 2011 series and I suppose it might be possible to put all the episodes from both series into a single order. (It would be tricky, though, because both shows use non-linear storytelling.) Furthermore, P4GA assumes you've seen the 2011 series and doesn't care about spoilers or about the plot. It assumes you already know and doesn't bother trying to fill you in. The first half of P4GA is a bunch of plot-free filler episodes with the gang hanging out together. What's going on? Nothing of importance, seemingly. Murder mysteries? People being thrown into the TV? Persona battles? You'd never guess any of all that was going on. Instead we get comedy antics like the gang performing live on stage or going shopping.
At the same time, though, events unfold differently in P4GA. There's a new character, Maria. There's also a subtle difference in that this is a New Game Plus, which apparently means starting a new game with all the goodies you won by completing the game on a previous run-through. Yuu here is thus stronger, braver and a better fighter than last time.
Did I like the show? Sort of, ish. I'd never recommend it and I can tell I'd have struggled if I'd never seen the 2011 series, but it's moderately entertaining. For a start, they've improved the show out of sight by ditching stuff that didn't work. The murder mystery investigation and weekly persona battles? Boring. Gone. I much prefer those filler episodes with everyone goofing around, which have their own kind of charm. It's still a fairly mediocre cast, with the girls in particular being mostly indistinguishable, but it's fun to watch them bouncing off each other and being friends.
Sometimes it's silly. Ep.2 is practically a character study of how much of a limp noodle Yuu can be, accepting twin invitations to hang out one afternoon. A sensible person would choose one of them, but not Yuu. (We are after all talking about the embodiment of a voiceless, anonymous, unseen video game protagonist whose name is "you" and whose eyes and colouring suggest a cardboard cut-out. Look into Yuu's eyes and, more often than not, there's absolutely nothing there.) Similarly ep.5 depends on idiot plotting. Our heroes need to draw a big crowd and so organise a live outdoor stage show with a famous retired super-idol... but then sell their tickets on a limited numbered basis, so that when extra people turn up, they're screwed.
I was also irritated by the start of ep.8, the Christmas episode, in which the show appears to turn into a harem anime. In fairness, though, this is the launchpad for one of the show's funniest and most inventive comedy episodes. If nothing else, you've got to laugh at Martial Arts Santa.
The filler episodes mostly entertained me. There's a character-based story in the emotional development of Marie. I also liked the existing characters better. Yuu is less pointless, while Kuma didn't make me want to beat people to death with bricks.
Halfway through, we suddenly get a two-parter about one of the original game's villains. That's a dark character study, showing us the world through the eyes of a bitter failure with no talent. He rejects friends, i.e. "the power of that thing you call a bond". After that, the Marie stuff gives the show a plot. Everything she'd been saying about making memories becomes significant and our heroes embark upon an unusual investigation. Marie's identity proves to be both interesting and complicated, then there's some moderately charming stuff towards the end with all the friends together again. The divine superpowers are funny (e.g. the weather forecast), but so are the Evil Valentine's Chocolates. It's a nice ending.
(Mind you, the OVA episode is a lot less nice. I presume it's an animated adaptation of one of the game's downer endings.)
I quite liked it. The story's messages are of interest, e.g. "It's human nature to wish for a lie". I liked Marie and I think her story is distinctive and, in the end, cool. Some of it's even worth rewatching, e.g. the more light-hearted stuff in last two episodes. The friendship theme is also mildly heartwarming.
If you've never seen anything Persona-related, though, run away hard and don't stop. Avoid this. Don't waste your time. There's far better out there, even if you're lucky enough to know all the background information required to follow what's going on. Its plot is basically barnacles on another plot that's being kept offstage. Yuu's still an anti-character. (I realise that's the point, but he is.) I like the cast better than they deserve and I'm particularly fond of Nanako, but you could still do better.
I'm still quite glad I saw it, though. It's interesting. I'm drawn to eccentricities.