Changing Secondaries and Getting New Skills

I was wondering, what's the best way to get new skills once I have ascended and want to change my secondary?

Can I do quests I've already done over again, but with my new secondary, to get the skills? Or do I need to do it with a different character?

It is efficient to just buy my skills and get new skills points via XP gain or is it best to start a new character?

Thanks for the advice.

I was told that you'd only be able to do the Class Specific quests, and you'd have to buy the other skills.

But, I havn't changed my class yet, so my advice has about the cost equivilence of a grain of salt.