So, for instance I don't know if the person you feed on would later become a vampire if not cured or if they are just someone you feed on. Mind shedding some light on Skyrim Vampires for me?

Oh wow... I made a huge fail on the title..

Anyone you feed off of will not turn into a vampire but that would be awesome if they would.

It would be awesome! But at one point you'd lose people to feed off of.

Nope. Just food for the player. They will not turn into vampires.

Then you move to morrowind or hammerfell and it starts again but if I remember correct only very old vampires can infect people they pick or just feed off them.

If you're interested ingame book http://www.imperial-library.info/content/immortal-blood is probably the best source about vampires in Elder Scrolls universe in general.
EDIT: Forgot to mention that the playable vampires in Skyrim are Cyrodiilic vampires, not Volkihar.

I cannot wait to meet the Volkihair vampires I just brushed up on Immoral Blood with the link above I am getting more and more excited to see Dawnguard.