If me and my friend start a new character that is the same class together, can we also do the personal story areas together too?

Yes.

 

Just make sure you have the option enabled in your preferences to allow other people in your story missions.

I think you need to do them twice, one for each person, since you're a spectator?

I believe there are some Phased Story areas where there can only be one of a given class, like the instance where you get your Saber in the JK and JC storylines.

 

But other than those you should be good to go.

If you and your friend are the same class, you either A) Do your class story missions twice together, OR do them once alone. There's no real reason to do class quests together, some things can be a challenge, but it's still meant to be done alone with your companion.

 

So you can just do your own class missions at the same time, or take twice as long and do them together.

 

To do them together, though, as someone has already pointed out, you have to enable having two of the same class in a class story area, in preferences under the social tab.