Here is a list of things I wished someone had told me, when I first started playing the game, and might be of help to new players. Feel free to post more tips and I will add them to the list.
1/ You don't need to do all of the side quests. There is plenty of XP available from the Main story quest to get most of your levels, you might just need to fill in the odd level here and there with dungeons/fates/levequests, but best to leave most of the side quests to help you level any future alternative classes. Side quests are standard gold colored quest icons, with no fancy bits around the edge.
2/ Do all quests you see with a blue quest icon with a + by the side. These will unlock all manner of things from dyeing equipment to opening up dungeons and raids. As you progress through the game, more blue+ quests will open up to you, so always keep an eye open for them. The only ones you might want to ignore, are the ones that say "So, you want to be a........" as they open up different adventure/crafting/gathering classes. Only take them if you want them.
3/ Follow the main story quest - gold icons with fancy bits around the edge. Most of the dungeons, mounts, equipment and game content is only opened up through following the main story, so that's a must do. Also don't forget to do your class quests. They are available every 5 levels, do head back to your class trainers every time you hit 5, 10, 15 etc. You can get some cool gear, decent XP and story through these.
So, to clarify: Do Main Story quests, Class quests and blue+ quests as a priority.
4/ When doing dungeons, there are some pretty cool youtube videos out there to help you out, and only last 2-3mins or so. Well worth checking out for some of the later dungeons. You'll find that alot of the mechanics in dungeon boss fights are similar in different dungeons. For example, if you get a gold flashing icon above your head in one dungeon, it'll probably do a similar thing if you got it in another dungeon. Keep an eye out for the stack-up icon, or the chains linking 2 players, they are quite common. Of course most of the fights have their own unique mechanics too :)
5/ Challenge log - this resets weekly and gives you whacking great XP bonuses for doing various activities so when you get access to this keep it in mind if you are needing to level up. The challenge log will get unlocked during the main story quest.
dungeons : When doing a dungeon for the first time, mention in chat that you are doing it for the first time. High level players will rush through a low level dungeon since it is very easy with maxed out gear. If they know you are new, they will probably go slower and be more forgiving if you mess up. They won't really mind since they get extra rewards when playing together with a new player who completes a dungeon for the first time.
Classquest : when above level 50, classquests are every 2 levels instead of every 5. Don't forget to do them.
Selling items : You can sell items on the market using Retainers (will unlock with main story), it's a great way to make some extra Gil. Don't forget to pick up your money after a sale ...*looks at Mac*, really don't forget that....
Macc Edit: Yes, I didn't realize I had to get my cash back from the retainer. I had nearly 2mil gil sitting there.
Glamours : Don't like the look of your gear. Good news, you can change it using Glamours (blue+ quest to unlock) *looks at Karofin* you don't have to look like a homeless guy if you don't want to.
As always if anyone needs any help with anything, just shout up in guild chat and we will happily assist.
Eat food every 2 hours (raisins are very cheep) - you get a 3% exp bonus. It may not sound much but it adds up
Always log out in a safe spot (there will be a moon symbol next to your exp bar). This will give you a rested exp boost.
Before you enter your first dungeon the is a building near by called "The Hall of the Novice" which is great basic training for your class role in groups and will rerward you with role decent gear.
Can't think of anymore now, but please post your tips here, and I'll add them to the list. Please also tell me if I'm talking rubbish about anything here, and needs changing 
Macc