Fresh 50 Warrior, what are the linkshells I need to join?

Just hit 50 as a Warrior. I have not done ANY endgame content yet.

What are the Hunt/dungeon/etc. Linkshells I'll need to join on this server?

You should get most of your help from your FC. If they are unwilling to help you out might be time to find a new FC. Otherwise as far as LS's go you should check out Just Raids, send Amber Grey a /t to get in. I also would recommend getting into a hunt LS, can get a lot of tomes & allied seals from A ranks, and those plus poetics from S ranks. When I went full out for hunts, I was getting about 800-1000 tomes & seals in a few hours or less. Look for Lanna's Hunts, Okami's Hunts, or Kirim K Ultra hunts for hunting(note, the first two will remove players who are offline for over 24 hours due to limited space)

Assuming you're not rich or a boss crafter who can make high ilvl gear, and you want to hold your own in endgame dungeons trials and Raids...

As a new-to-50 tank, your immediate concern will be enmity management; no matter how skilled you are, an ilvl 115 DPS that knows what they're doing will pull aggro off if an ilvl 60 tank. Weapon Damage is a huge contributor to your aggro build and (excluding Atma Weapon)is directly tied to the weapons ilvl. With that in mind... I recommend working on Hunts while you work on acquiring your Barbarian's Bardiche and 2 Battle Dance III materia. These are for your relic Weapon, and can be used as your weapon for the moment. The materia is the tricky part and can be expensive, though sometimes the cheapest way is to buy the weapon with the 2 materia already melded. Use Hunts to reach a MINIMUM ilvl of 52 (hunts are the fastest way to gear up). At ilvl 52 you can complete that relic quest to get your ilvl 80 weapon and upgrade it to 90 with some tomes gained from hunts. You now have a reasonable weapon for dungeoning with! Yay :D Keep in mind at ilvl 52 you'll be more or less carried thru the relic quest (though that'll probably be the case your first time thru anyways..) I'd recommend something more akin to 60+. If you need any help acquiring your relic, send me a /tell in game, I can help make it quicker/cheaper, and probably get you a full party to run they it, me and my friends are a very newbie-friendly bunch.

Continue to work on hunts, and whenever you're up for it, start the Legacy of Allag quest found in Mor Dhona. This quest ultimately unlocks Labaryhth of the Ancients (aka CT) which I recommend bringing an experienced tank with you to complete. Once you know that fight you can easily farm ilvl 80 gear as it's not tough but it does have some tank mechanics that need to be learned. Completing that allows you to unlock Syrcus Tower, and this is important because you can quickly farm ilvl 100 gear, Sands and Oils of time to upgrade your tome-bought gear from 100 to 110, and a UAT which (at the cost of tomes) can get you your ilvl 100 weapon, upgradable to 110 with a Sands of Time, which is what most people are currently using. This fight is easier on tanks than the first, and can be farmed in DF.

As for the actual question that I totally steered away from... I can invite you to Coil Learning LS which is a good place to get people to join you on runs. While it is intended for Coil (endgame raid) it's not unusual to use it for Primals, CT or ST and in my experience the people in there have good attitudes. Other than that, get yourself a good Hunt LS. :)

I say do the Labyrinth and Tower missions to have some fun and maybe earn a nice piece of gear, but focus on the hunts to gear up. You can buy all the ilvl 100 gear you need from the tomes & seals you earn doing the hunts, and since each A rank gives you tomes & seals it goes twice as fast. If you're in one of those hunt ls's(okami is probably the best) and you show up to each hunt and join a good pt, you could probably hit ilvl 100 in one night(4 to 5 hours). You will be surprised by how fast you can get tomes & seals. I know when my FC was running hunts non-stop, we would get about 1000 tomes and 1000 seals every 60-90 minutes. Once you've got your gear up, join the learning LS matt mentioned, and most importantly listen to the advice given, too many people think they know how to play this game and refuse to change, then leave their party wishing they had a different tank.

Oh and matt, any dps "that knows what they're doing" knows to back off when they start to steal aggro, its not hard to keep an eye on your aggro level. Sometimes the best thing you can do if your tank is underpowered, is put your weapon away for a second or two, you do far more damage when you're alive and put less strain on your healers. Non-tanks can help the tank be more effective by letting the tank build some aggro, too many(specially bards) jump in before the tank even gets a single hit in.

I won't steal aggro and if I do so accidentally I release it (unless it's about dead). I agree 100% that a DPS that knows what they're doing will back off. However, the tank should keep in mind that a DPS who's not allowed to DPS within reason is likely to not enjoy the run, which isn't fair either.

It's one thing to wait for the tank to establish aggro, but when you have to sabotage your own rotations too often it ruins the experience. This is why I encourage tanks to gear up and get a nice weapon. Something I learned maining PLD is that tanks are in direct competition with their party, because the DPS is going to want to do their job just as much as I want to do mine.