For someone like me who doesn't own Persona 3 or 4, Persona Q is a decent replacement. Let's start with the good. There's a lot of good character interaction between the P3 and P4 cast and being able to mix up your own parties and such is really cool (at least at face value, but I'll go into more detail later).
This game is probably most noteworthy for being a dungeon crawler. Some people might not be a fan of that kind of thing, but I personally enjoyed it (though admittedly the final dungeon was kinda annoying). You have a map on the bottom screen you can draw as you go along and you can mark points of interest with notes and such to come back to them. Each dungeon has a unique set of FOEs (short for Fysis Oikein Eidolon), which are basically really strong monsters that live in the dungeons. Fortunately they're visible on the map and they only move when you do, so you don't really have to worry about accidentally encountering one. For the most part these FOEs match the dungeon they're in and the dungeons are designed pretty well around them.
Now the bad. The main problem with the game is its battle system. In case you don't know, you equip a "sub-Persona" that will give you certain boosts in HP and SP that increase your max stats during battle. This means if you take damage less than or equal to said boosts, you ultimately end up taking no damage at all after the battle is over. Think of it as like a shield. Same thing applies for SP, if you only use a small amount you basically end up not using any at all. Not only that, but if you hit an enemy with an attack it's weak to, you get to move first next turn AND your next spell counts no SP. Sounds cool, right?
It is, until you find out that every single spell costs an ♥♥♥ load of SP. Here's a comparison for you. In Persona 4, a simple Agi spell costs 4 SP (Agi is the fire spell of the game). Considering your main character starts at about 40 SP, that's not bad. Switch to Persona Q, that cost has been doubled to 8. At first glance that doesn't seem too bad, until you realize said main character now only starts with a measly 24 SP! So not only have spell costs been doubled, your SP capacity has also been basically halved.
Physical attacks do chump damage as well, so the game pretty much boils down to finding an enemy weakness and praying you don't run out of HP or SP to exploit said weaknesses. See, there's a catch to the whole "free magic" mechanic: if you get hit by any attack, you lose that "free magic" boost. So imagine this: you hit an enemy's weak point and get boosted. Then they immediately recover and slap you across the face, completely negating your boost. Now you're pretty much screwed since you've used your "shield" for that first spell and now you have to dig into your already meager SP supply to try again, only to have the process rinse and repeat.
One more thing of note. There's a standard in Persona games where the protagonist can freely switch their Persona to change what kind of abilities they can use in battle. It makes your main character really flexible and can save your hide in a lot of situations. Persona Q throws that out the window in favor of the whole sub-Persona mechanic. This mechanic does spread to all your characters, but you can't switch Personas in-battle, so that pretty much throws out any kind of flexibility you would've had.
But I'm not done. There's something bigger, something even worse: the freaking final boss. By which I mean the final final boss, as this game has a final boss gauntlet. That in itself is not a bad thing, in fact I quite enjoy the first of said gauntlet and would've been a great final boss in my opinion (if you've played the game I'm talking about the spider thing). The problem lies within the last boss of the gauntlet. Spoiler tags ahoy.
The problem with the last final boss is it has an attack called Gears of Time. It basically inflicts your entire party with a "Doom" status. There's a countdown that decreases every turn and once it hits zero: immediate death, no matter how much health the character has. The game gives you a hint that you can use reviving items/spells, but it's basically useless because literally all they do is increase the countdown by ONE turn. The one good catch is that the countdowns are staggered, so only one person will die each turn. What this means is basically you have to load up on revival items or have almost everyone know a revival spell so that when one person dies they can be immediately revived next turn.
To top it all off, the highest difficulty mode has a catch where if your main character dies, instant game over. No surprise, because that's a Persona standard. This means if you're on said difficulty, you pretty much have to assign one character to be constantly giving the main character revival items/spells so you don't game over, on top of just trying to survive said onslaught. Thankfully said mechanic is limited to only the hardest difficulty otherwise the boss would literally be a nightmare. Already I consider it one of the worst final bosses in terms of the mechanics of it, but that makes it even worse.
The plot? It's just a throw-away, an excuse to mix the two games' characters. It does have a plot twist or two, but overall it's nothing to write home about.
The last thing I have to say is less of a problem, but still noteworthy. At the beginning of the game you can choose between playing as the P3 or P4 protagonist and their respective casts. Taking this as-is it's a nice option, it can add to the replay value of the game. Except in this case, it really doesn't. Speaking from experience here, playing as the P4 protagonist seems like the "intended" choice, and the P3 option feels really... rough, for lack of a better word. There's a few story plot points that support this, the main one being that the whole setting is inside an alternate version of the P4 school. I mean come on, you can't get any more obvious than that. Plus, you miss out on a hilarious event by playing P3 (if you're confused, it's the bathroom event in the third dungeon, it only happens if you play P4).
tl;dr Persona Q had a lot of potential in mixing the casts of P3 and P4, but the developers focused too much on the crossover aspect with Etrian Odyssey series and ended up ruining the battle mechanics. If you can deal with a very flawed battle system, you'll experience some good (and mostly hilarious) interactions between the characters. It's a decent game if you can get it on sale for cheap, but don't spend a bunch of money on it.