As many stated before, the game has a very slow start and is heavily story focused. While leveling, you rarely NEED a group but there are several dungeons and trials along the way that can only be done in a group. IMO the leveling is far more group oriented than games like WoW since during the leveling process you don't ever NEED to do a dungeon or be in a group ever, but doing the zone quests may lead you to one, but they're not required by any means.
Due to the long 2.5 second GCD, and classes only having a few skills at low levels, combat is very slow until about level 30. For faster paced combat try Pugilist or Thief, which evolve into Monk and Ninja respectively. That said, all classes really start filling all the downtime by level 50 (not much of a consolation I know since that's a long way off, but it is what it is).
The end game is really where you'll obviously need groups as dungeons, trials and raids all require groups.
The game does now also offer two "skip" potions: a level jump potion for a single class that will jump that specific class to level 60 and can currently only be used once per account and two story skip potions; one for just A Realm Reborn content and one for both ARR and Heavensward content so you can jump right into Stormblood. They are separate. Buying a level boost will not complete any story quests (which are required to be completed to unlock the majority of the games content) and buying a story skip will not level you. The Job Level Boost potion is $25 and the story skip potions are $18 and $25.
HERE is some information on them from the reveal and HERE is an article with the prices listed for the potions. Additionally, HERE is a link to information on Realm Transfer, which costs $18.
Unfortunately if you want onto Gilgamesh you'll likely need to pay to transfer onto that server as with it being such a high population server I highly doubt the character creation restriction will ever be lifted.