I just started P3P this weekend and only have an hour or so logged.  I've never played any Persona game before (I don't really count watching the Endurance Run).  Does anybody have some advice on things I might miss or do wrong along the way that could come back to bite me later?  Since it's my first playthrough, please try to avoid spoilers.  I'd rather miss out on an advantage than have the plot ruined.  

• You can social link with various characters in the game. Each character is associated with a certain arcana and if you have a persona on you that corresponds to the same arcana, you'll get extra social link points. This makes progressing through s.links faster which is important since you have limited time.
• Try to finish battles as quickly as you can. Do this by exploiting the shadow's weaknesses and try to finish off each battle with an All Out Attack when possible. This is the fastest and safest method since battles can turn ugly pretty quickly if you're not careful.
• Always fuse new personae on the Velvet Room when you have the chance. If you get too comfortable with a handful of personae you'll find later enemies to be harder.
• Later on in the game you'll get a Compendium in the Velvet Room where you can store and summon previously registered personae at a price. This is really useful for fusing later on. Make sure to register new personae that aren't in the compendium. Newly fused personae are automatically registered to the compendium.
• Try to do well in school since it'll reward you with card sets for your personae and it improves your Charm in the process. So try to work on your Academics when you can.
• Other social links won't unlock until you have your stats (Courage, Academics and Charm) at certain levels so again, try to work on those in the evening at Paulownia Mall.
• It's highly unlikely that you'll get every social link at max whilst working on your stats. So work on your stats first so on your second playthrough you can concentrate on social links instead.

The arcade is your friend, focus on leveling up you stats there or at the Cafe at night, try not to work on stats during the day since that is when most Social Links occur. 
 
Also try to save non-school SLs for days when you don't have school or no one is around because of exams, and there are plenty of those days. 
 
If doing the female side make sure to do team member SLs first.

It's Ok to tell friends that phone you that you are busy. There is one SL that is Sunday's only, so make sure you have enough Sunday's available to finish it and the days off only SL. You'll almost always get a call Sunday asking if you want to hang out, but if you do hang out too much you won't have the days left to complete the SL.

If your like me and don't want to get 100% of everything you'll probably not be following a guide the whole time which is the absolute wrong thing to do if you want to enjoy the game.  I suggest though that you should try to get as far into tartarus as you can in one night so you have time to study or do night social links.  When your in the dungeon though don't get cocky and when you get to a floor with mini bosses take the teleportation device so you can save.  Save often too cause you may find yourself under leveled. 

Go into Tartarus more than you think you should, when you get to the first mini-boss in a block go down a few floors and grind a little. I didn't do this and wasn't ready for time-specific events, even on Beginner.

Thanks everybody, that is exactly the kind of advice I was looking for.  The last thing I wanted to do was look at a guide, but I also wanted to ensure I didn't make uninformed choices and end up screwed 40 hours in. 

Hmmm, some differences of opinion on how much one should grind.
 
My opinion is that when you are going through Tartarus, you should go through each floor and fight every enemy you find (unless you've explored the whole floor and they're reappearing, they will do that after a few minutes, then you can avoid them on your way back to the stairs).  If you do that, you should be pretty well covered and not under-leveled -- especially if you take XP bonuses from Shuffle Time.  On a first playthrough I generally like to go to Tartarus either once per week or once every two weeks.  You may also want to wait for a call from Elizabeth/Theodore about missing people in Tartarus to go in for the first time in order to avoid backtracking a bunch (though I wouldn't rely on that because in some months he doesn't call you until quite late).
 
Shuffle Time is different in P3 than in P4 (which I'm assuming you've seen from Edurance Run).  You can get weapons, money, XP bonuses and healing as well as new personas from it.  You should definitely use this to your advantage, especially if you want to avoid being underleveled.  Here's what I'd recommend:
 
1.  If you see a Persona in shuffle that you don't have yet, take it first.  Even if you aren't going to use it, you can always take it to the Velvet Room and fuse it to make something more interesting (note that any persona that you add to your lineup through shuffle time will be automatically registered in the compendium if you haven't already fused it).  If you run out of Persona slots, you can always replace the new ones you find with other basic ones in your current lineup and buy them back later from the compendium if you need them for something.
2.  If you have the personas in the shuffle already, take XP bonuses.  This will decrease your need to grind considerably.
3.  If your protagonist needs some healing, take the cup cards -- there's a chance it'll heal your whole party, so it's useful if you really need it.  Though I find that the clock and your items/skills are better for this.  It's okay in a pinch, though.
4.  After you've got the available personas and you feel like you're not getting all that much XP from your battles (or enemies are running away from you), take the weapons or the money.  The weapons are not the current best weapons you can get, but loot is loot, and you'll need all the money you can get.
 
For the life of me, I've never been able to figure out exactly what conditions you need to meet to get shuffles, but I know for sure that you will get a shuffle when you finish a battle with an All-Out attack, or when your protagonist lands the final blow in a battle.  You'll sometimes get them in other situations, but I don't know if they're just random or if there are other conditions for them.
 
Also, one last battle tip.  Do not lounge around on one floor for too long -- you cannot just sit on one floor, wait for enemies to reappear, and just grind a bunch in one place.  Bad things will happen ;D  Try it out once just to see what happens, but you'll probably want to avoid it until close to the end of the game XD
 
Whew, long post is long, sorry for all the rambles D:  Hope that's helpful, anyway.

Another tip, switch your characters to manual control (much easier I admit though i still love my AI control for some reason) but it'll be faster battles for you in the long run.
 
I guess in the end chose wisely who you SLink with and how much time you spend on things. Also know that NG+ you keep all your maxed Charm courage and so on including level (i've just found that out myself~!) so even if you can't get all the Slinks do what you can and it'll be easier during round two. XD