I'm new to MegaTen games, new to Persona games and I have a question...do I need to to go into Tartarus every time it lets me? Just when I'm healthy? How often?

I love the game, love the writing and I'm looking forward to picking up Persona 4 this fall.

You should always have an elizabeth quest that asks you to get something from the topmost floor you can get to.  Make sure you go to tartarus enough to be able to accomplish that within the alloted time without avoiding most fights, and you should be good to go.  Also yes, try to go only when you and at least 3 others are at normal condition.

It's been a while since I've played it but you should try to get as far as possible in Tartarus up front every month. Like make it to the end of a section where it won't let you go further. Whenever boss floors come up you should probably use the teleporter if you're able then save then go back up. Boss floors are the only floor you can teleport to if I remember correctly so don't use teleports on a floor if you don't need to. You should go in only if your party is healthy since it has detrimental effects on the party's performance. You can use the restroom to improve your health. You should spend the rest of your time working on your character stats like knowledge or your S-Ranks once you've done a section or when your party members are sick/tired.

I would pretty much clear out an entire block right as they opened up. You don't need to go right away, but there are some timed objectives a little later in the game. The timed objectives aren't required, and you always have an idea of when to go do them.

I always went when I was healthy, I believe it affects your stats when a character is tired, and being ill can spread a disease around your party. I never needed to grind to beat the game, I just cleared each block once.

Yeah, only go when you and your party members are healthy. Going in while sick only invites trouble. Don't stress over it; just try to enter Tartarus as often as possible and make the most of each excursion. Do it right and you'll end up being ahead of the level curve at all times without having to grind much.

NOOOOOOO

You don't have to do it every night. Just make sure you've cleared the entire block (including boss) before the full moon.