Yes, you can skip all the bullshit side quests entirely when leveling-up and do only the stories. It goes even faster if you drop $30 on it for a 60-day sub. I finished my Sith Sorcerer last year that way and got through about half of the Republic Trooper before apathy took me again.
The past cannot be changed. The future is yet within your power.
I tried this game because they gave out a level boost on Twitch Prime and then made a new character instead. I made it off the first planet to some big empty spaceship wandered around for half an hour and couldn't find my class trainer but found a million vendors with items I didn't need, googled for advice and still couldn't find a trainer, and uninstalled. Not sure if that's my fault or the game's.