I was also my adviser's first PhD student, and I got my own first student recently. The short answer to this question is that it's not obvious what to expect and it depends a lot on the personalities and academic upbringing of both student and adviser.
As a PhD student I had a lot of trouble with my adviser's micromanaging. I didn't like that, but he could do it because I was his only student. Second problem was that he had no idea what my thesis should be about. I tried three different projects until we finally figured out what to do together. Grant money was not a problem because he had both a grant and startup money, so I was research assistant for the whole duration of my PhD. Once we started to get along, after two years of working together, things came out nicely and by the end of my PhD I had a few good papers to show.
From my perspective as a guy starting to advise younger scientists, things are different. At first, I tried not to repeat things my PhD adviser and my postdoc bosses did, and I perceived them to be wrong. Then, I realized they may not be wrong if applied to other people than me. The same thing can be said about things I thought my supervisors did right.
So, first thing to expect from a first time adviser is to try to adapt what he perceived as good in the groups he worked before, while avoiding what he perceived as bad. If it doesn't work, they might try to overcompensate doing the opposite, so they would look inconsistent. For example, in my case, I had lots of trouble as a student approaching completely new problems. I'd try to read entire books and after weeks of disorganized searching, I'd end up with no answer. My adviser original approach was to let me do my thing which was his adviser's approach and worked for him. When he realized it didn't work for me, he started micromanaging things very tightly. Though the extra discipline did help, I resented it.
Another thing that might happen is that the new adviser might be inexperienced with teaching. This might be the case of people who excelled at research and got their position because of it. As their student, you may expect them not to be very pedagogical when teaching you new things if they actually do.
Many people who become new faculty feel insecure about their research. Their older colleagues may have hundreds of publications, and they may have a few tens at most. Plus, they need to get tenure. Some new faculty tend to get abusive with their students, plus they keep jumping from one project idea to another and it's very frustrating for a new student to spend three weeks researching something only to find out it's not needed anymore by the time they have an answer.
Something else I've seen in graduate school is new faculty who don't get tenure and their students need to finish the PhD with someone else. You can't predict if a new faculty will get tenure right away, but, in my field, if they don't make new collaborations, they most likely won't publish enough to get it.
I don't think it's inherently bad to be someone's first PhD student. They are pressed to obtain funding as a tenure requirement, and the ones that are good at grant writing, will soon have a larger team. As the first member of that team, your research assistant support will be ensured.