I'm in a situation that I'd love to hear some second opinions on. I am a second semester senior at a well known engineering university. I was never interested in going to grad school during my first three years of undergrad. After an internship, I realized that I might want to go to grad school but didn't apply to any by the deadlines because I was hoping to get a job in industry. When no opportunities panned out, I approached a professor to do undergrad research in his group this semester in the hopes of getting a recommendation letter for future applications. Once we talked, he mentioned that he had a PhD position available and asked me if I would be interested. Once I learned the details (the funding was coming from an external company that I would have a good chance of getting a fellowship from), I was extremely interested and said that I would join his group to pursue my PhD.
Now that I have accepted the offer, I am starting to have reservations. I believe that I would love the research and the coursework, but I'm not sure that I want to stay at my undergrad university for that long. Would it be beneficial to go somewhere else? If so, how should I go about considering other universities? How should I let my potential advisor know about the doubts I am having? Any insight more experienced people could provide would be extremely helpful!