Depending on local laws, you could just say "I have a doctor's appointment on day X, how should I proceed?". Work laws, in general, grant you the right to go to a medical exam/consultation without requiring permission from your manager/company. But maybe you'll need to hand a doctor's notice that confirms you were indeed seeing a healthcare professional. Even if the law is not on your side, any reasonable employer will understand the situation.
The same goes for having to sort out a banking issue, testifying in court, and so on. As long as you can explain the case and it is neither leisure nor an interview for a different job, it should be no problem, people have these issues all the time (except for the court case, I hope). If you are not comfortable, you can at first ask generically "How should proceed if I had an appointment during work hours?",
The theoretical question is best posed on Human Resources (HR), while the practical question should be addressed at your direct manager, which will be known to you no later than on your first day.
If however, this is some gray area case, like accompanying your grandmother on a health check, or some already postponed meeting with the principal of your daughter's school, then think well how often an event like this would happen. If it is way too usual, I think it becomes basically a requirement for you that you should work at a place that accommodates this. If these events are to happen once in a blue moon, then explain to your boss that you have this situation and that it is not a recurring thing so that an exception can be made for you, but you both understand that exceptions will not take place every month.
Please refrain from asking for a day off if you are doing something that looks frivolous, even if it is something very important for you (at least if it is not as important as your new job). Examples that come to mind and would put a bad light on you are "I wanted to go to this author's book signing session", "I had this laser tag game with friends scheduled since last month!", "I'm going to this chess championship I have no chance of winning".