It's really not too difficult.
From inside the city of Limsa Lominsa, lower deck, there's a port. You go in the port and wait for the ferry to arrive, then you get a boat ride (takes a few minutes, bring some crafting if you don't wanna get bored). The boat ride will drop you off at the far western side of Thanalan, at Vesper Bay. There's a road that leads out of the port; stick to the road and head east. You'll pass Camp Horizon (tag the crystal on your way). East of Horizon, following the road, you'll go up a couple of ramps, and then there's a small area with antlions that will one-shot a low-level character; it's dangerous if you're not paying attention, but so long as you don't wander too near them, you ought to be okay.
Keep going east on the road, and you'll get to a fork. Camp Black Brush is right nearby (tag the crystal). You can divert south to get to the city of Ul'dah. The crystal plaza in Ul'dah is to the west, should be easy to find on the map once you're in town. Tag the crystal, then go back to the road headed north.
Once you're back at Camp Black Brush, head east again. There will be another small segment where you have to watch out for antlions, but again, you have plenty of room to avoid them - just don't space out while making the trip until you know where they show up. You'll eventually come to Camp Drybone (tag the crystal). If you keep heading east, you'll have a long and winding path up into the mountain area, but there aren't really any aggressive mobs along the way.
Eventually, if you stick to the road, you'll zone into The Black Shroud. Keep following the road north. You'll eventually pass by Camp Tranquil (tag the crystal). Not really a lot of aggressive mobs in this zone if you stay by the road, but watch out when crossing rivers - I got nailed by a fish for 1200 on my first trip. Following the road will eventually take you to Camp Darned If I Recall What It's Called. At that point you're pretty close to Gridania and you ought to be able to see how to get there from the map - the road doesn't lead right into town, but it meanders by both exits from the town.
The crystal plaza for Gridania's to the west a little, should be easy to spot and get to.
The whole trip will take you about an hour and a half, counting ferry time, not counting dying along the way. I mean it about tagging the crystal - not only does it mark the crystal so that you can teleport there, it sets your home point there, so if you get munched by something along the way, you only lose a couple minutes of travel instead of half an hour.
Once you've tagged a crystal, you can teleport there at any time for a cost of 6 anima, or 4 if you're already in the zone... or if you're going to the last crystal you tagged, you can "return" for only two anima. Your character starts with 100 anima and will regenerate six anima per day. There's no way to speed up the regeneration (also no way to ***** up and not get it; you will get 6 anima per day every day, about 1 every 4 hours.)
SE has announced plans to reduce the cost of "return" to 1 anima, and to let your character designate up to three "favorite" destinations which will only cost 3 anima to teleport to. Probably you'd want to designate each of the cities, though if there's one city you can't stand, maybe not?
If you're in a hurry, a buddy can teleport you. If you're in a party with someone, and one person teleports, all his party mates ride free, even if you couldn't teleport to that destination yourself. If someone ports you, make SURE to tag the crystal so that you can port yourself next time! Also, tip generously - that person just spent an entire day's anima to save you an hour of running.