I want to find a cure for being a werewolf and then find the way to become a vampire. Do I have have to be out in the bad part of town and wait to get bit or is there a way to provoke a vampire to turn me into one?

Find some, fight them and run away.

I took out a whole cave of vampires but they never bit me or anything.

In past games I could get Porphyric Hemophilia by repeatedly activating a corpse of a vampire I just killed. Not sure on this one.

As for being both, once you are a Werewolf, your immune to diseases, which would stop you from being a Vampire. If you are a Vampire then become a Werewolf, then you are cured of Vampirism.

So, you can only be one or the other, not both in this game.

To cure your Werewolf, complete the Companions questline.
WARNING: PLOT SPOILER
Talk to Farkas or Vilkas or any Circle member, and they will want you to help them collect a Glenmoril Witch head and take it to Ysgramor's Tomb to have them cured. Collect two, and cure yourself as well.

Dang I finished that quest but I only grabbed one witch head.

You can go back and get the other witch heads I'm pretty sure.

Also, for contracting vampirism, I just go to a known vampire lair (Bloodfang Cave in Whiterun, Haemar's Shame in the Rift, etc.,) take off all my armor, turn the setting down to Novice, and let some vampires wail on me. Usually works faster than hoping they spread it to me otherwise. Also, apparently when they use Vampiric Drain it ups the chance of your contracting it so there you go. But be careful since even at that low level, a lot of vampires or a Master/Volkihar vampire can beat the life out of you.