I am a Web developer and I am working with a relatively new company, I am hired part time because I'm a student. A few weeks ago the company hired another developer for full time that wants to do some changes and the bosses are green light about it. So here comes my problem. I asked him face to face what should I learn to be useful and he told me "X" would be good since it helps the project to be delivered faster and of course I started to get a hang on it and the bosses asked me how I was doing and what am I studying and I told them that I studied "X", because I talked with my co-worker and we agreed upon this. At that moment they asked him about this and he denied this and said that he never told me as such. How can I solve this issue without quitting or looking like I am at fault.
Just shrug it off and keep doing the best you can. It doesn't really matter unless it's impacting on your reports.
Next time ask via email so you have a paper trail. It's important to keep trails on everything, something as simple as:
"Hello [insert name], just clarifying what I should focus on, it was XXX wasn't it?
Since time has now passed since that conversation, it's increasingly difficult to "solve" this one.
A response would have been "Well, I must have misunderstood you - so what did you tell me?".
All you can really do is move forward from here. Over time, you'll gain more experience in assessing whether what you're being told is appropriate or not.