Seeing as that I'm dating someone who plays oboe, I never hear the end of both of these topics

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First of all, with vibrato, oboe is by far one of the toughest instruments to get it right with. In my opinion, the oboe needs to have enough vibrato to not sound stiff, but not so much that you sound like you are shaking. It's all in the pulsing of the air (Or so I hear). But, regardless, a teacher is the best resource you've got and can tell you so much more than I can.
In terms of oboe pieces, there are really some great things out there, mostly baroque.
Any of the Telemann oboe concertos or the Albinoni oboe concertos are good (I'm particularly fond of the Albinoni in D minor).
Marcello also wrote some fantastic oboe concertos, again the one in D minor is one of my favorites.
Go on to youtube, or google and type in these concertos, listen to them and figure out if you'd like to give them a shot. If you find one you like, order on sheet music plus, or other various sheet music sites.
Hope I could help.
-corey