I was kinda doing that for a while with some friends. Only kinda, because I was playing a new class on my existing character, rather than an entirely new character. So I didn't get to participate in the quests and cutscenes. Still, that was enough to get the idea that FFXIV is not designed for that kind of play. Conversations and cutscenes don't acknowledge others in your party at all. World enemies are even more of a cakewalk than they are when playing solo. And there's semi-frequent solo battles which force you to break up the party.

If you want an MMORPG which really can be played as a group from start to finish, I recommend Star Wars: The Old Republic. It's designed as a multiplayer experience; all characters in your party will appear in cutscenes, and aside from class quests, everyone gets a chance speak or choose paths of action too. There's a lot of speaking for the PCs, unlike FFXIV which gives you maybe one line every five conversations. And it's all voiced. You even get to help each other in class quest instances. I recommend picking different classes though, otherwise you must do the same class quests multiple times. World enemies are not any more difficult when playing as a party, but the game does feature heroic quests which are designed to be done with two or four players (two-player heroics generally can be done solo if you're careful, take your time, and utilize your companion well).