PhD student I am working with as an undergrad may pursue something similar to what my senior thesis is on, is this a conflict of interest? I began researching for my thesis on subject X several months ago, and the thesis program is starting in a few weeks.
I did not tell the grad student at the old uni about this yet - because originally we were working on different things, his idea seemed cool, and naturally, I want my name on as many papers as possible. However, my adviser told me that my research idea may be enough to be considered a significant contribution in the field. A lot of people in research are interested in how X can be applied to different areas right now, but my idea on how to apply it is completely my own and unique.
I feel I should either only work on my thesis at my new uni, or email the grad student at my old one, let him know I am researching X right now, and that if he involves X in his research, that might not look good for me.
How should I go about this? I feel bad because the grad student invested time into teaching me for the last 4 weeks and getting me caught up, but I also want to do what's best for me - which is 100% pursuing my research idea.
The fact that my adviser thinks it is publishable and may make an impact in the field is rare enough in undergrad, while the grad student I am working with is not sure what exactly he's researching yet - and I fear that he will look into using X now, which might cross with my research.
No.  One of the criteria for a problematic conflict of interest is that the conflicted person has power.  The PhD student does not have any power in this situation.
In general, collaboration is good.