There are two major exceptions: the fox who lives at the local shrine and Margaret, who lives in the Velvet Room. The fox's quests usually require a bit of legwork on your part because they're worded in very vague terms. You'll know, for example, that somewhere out there, a young boy wants more friends, but where he is and how you can help are things you'll have to figure out on your own. It's a lot of work compared to other quests, but completing fox missions not only bumps up his Social Link but also gets you discounts on his in-dungeon healing services. Margaret, meanwhile, wants you to show off your fusion skills by presenting her with Personas that have specific abilities. As with the fox, these missions are kind of a pain, but completing them bumps up Margaret's Social Link and the reward for maxing out is an immensely powerful Persona.
3. Finish dungeons as soon as you can. The story in Persona 4 is very closely tied to the calendar, and if you don't complete each dungeon by a certain date, you'll fail . Actions, like raiding a dungeon or studying in the library, take up time, and you only have so much time in a day to get stuff done. You typically have a few weeks to clear a dungeon once it opens, but don't dilly-dally. Start your exploration as soon as you can, keeping a close eye on the weather report. Several days of fog followed by rain is your signal that you're running out of time, but the game will pretty much club you over the head with the fact that your days are literally numbered, so you're not likely to just plain miss the important day. You actually have plenty of time to make it to the boss, but it's easy to get distracted by all the other things you can do with your time and suddenly find yourself having to rush through levels. You don't necessarily have to just plow straight through as soon as a dungeon opens up, though; you get better loot on rainy days, so try to arrange your schedule so that boss fights happen when it's wet out.
After you've cleared a dungeon, it will reopen with a new optional boss at the end of it. Not only is redoing the dungeon a great way to grind in preparation for the next big battle, but you'll also get a piece of swank gear for beating the optional boss.
4. Become well-rounded. Raiding dungeons will improve your combat stats, but other aspects of your personality need attention, too: Understanding, Knowledge, Courage, Expression, and Diligence. Different activities strengthen different personality traits - making it to basketball practice will boost your Diligence, while playing with a stray cat can bolster your Understanding, for example. Certain characters won't give you the time of day until you've raised a certain stat to a certain level, so you're going to want to develop them all. The most important ones to go for first are, arguably, Understanding, Knowledge, and Expression, because they're the ones that open up Social Links the most quickly. All five traits will offer up benefits of one kind or another, but those are the three that are most beneficial early on in your adventure.
5. Get a job. You mean catching a murderer, socializing with friends, looking after your baby cousin and getting good grades isn't enough? Nope. Ok, you don't actually have to get a job, but you're probably going to want to. Most of them earn you money (although your dungeon exploration should keep you pretty flush), but more importantly, they also open up new Social Links and often increase a particular personality trait. Working at the scary hospital will improve your Courage, doing translations at home will boost your Expression and folding paper cranes increases Understanding. The game will, fortunately, remind you when you're scheduled to work, though it's up to you whether or not you actually want to go. You can have several jobs at once without peril of losing any of them, so you can squeeze in a whole bunch of employment in between trips to dungeons.
This doesn't even begin to scratch the surface of a proper Persona 4 strategy, but it should at least get you pointed in the right direction. Persona fans, please feel free to offer your own suggestions for success in the comments, but please keep them as spoiler-free as possible. We'll let the new folks discover the joys of Void Quest all on their own.