It is possible to close that gap in a few hours by doing Fates in Northern Thanalan or Dzamael/Aurum but I would really not suggest this for your first class. You're gonna do this many times over to level your other classes down the road. I'd suggest just following your MSQ and enjoy the game at your own pace, there's no reason to rush it. You can always benefit from using food while you are looking at an extensive time of killing things since they give a +3% exp buff. Might not sound much but it builds up. Also make sure you log off or afk in areas branded as sanctuaries(pretty much anywhere close to an Aetheryte) so you get the rested exp bonus.

As for the buffs mentioned above I'm assuming that is referring to the Free Company buff, Heat of Battle, which also offers an exp bonus from anything killed as well as the food buff.