previously leveled up other classes before turning to a tank. By the time I got by PLD to Stone Vigil, I had forgotten the dungeon. Yes, it is very stressful to enter a dungeon where you don't know what's coming.
Generally you need to have equipment at around the level of the dungeon. The tank usually needs to be better equipped than the rest of the party. Having a better weapon allows you to hold aggro better.

Some people are elitist and will give you uphill for your equipment. This is wrong as you don't need to be that heavily equipped for stuff until you start hitting the lv50 dungeons.

The DPS classes often forget that it is also their job to manage their aggro. If you're losing aggro because they're attacking the wrong things, or because there is movement involved and the BRD is going psycho and not watching the aggro bar, then that is their fault, not yours. If you lose aggro to the healer, then you need to either change your rotation, get better equipment, or tell the healer to stop over-healing.

Are you part of a FC? Best bet is to find a healer lazing around there who is happy to run the dungeons with you. When I ran AK for the first time it was with a healer from my FC. They popped up at the start and told the DPS not to go insane, just take it easy since I was new. Some DPS may be unhappy that you're not going insane and speed running stuff, but most will be okay with it. Everyone knows that you need to learn somewhere.

TLR - it's not just about equipment.

For equipping, it depends on how much cash you have. Since you're a new tank, I presume very little. Do the story roulette every day and get Darklight gear asap. Get a lv50 weapon from your company - you won't be spending marks on much else. As soon as you can, start running the lv50 dungeons - preferably with understanding people that don't expect speed runs.
While CT is good for getting armour, the DF people are often unforgiving. As a tank, Ifrit HM is an easy fight. Run that until you get your weapon.