'I have to get up early' is a normal thing to say. It means that getting up early is necessary, similar to 'I must get up early' or 'I need to get up early'. This could be something that you have decided for yourself. 'I am to get up early' is a little strange. It suggests that this is part of a schedule which someone else has decided for you. It would be very unusual to use this particular phrasing in this context. The construction [be] + [to] + [infinitive] for a fixed future arrangement is quite formal. It has a very old-fashioned sound to it, and these days tends to be used mainly in journalism for ceremonies, official state visits and suchlike - not for ordinary people talking about getting up in the morning.

From the Oxford English Dictionary ( https://en.oxforddictionaries.com/definition/have ), this meaning of "have to" is: "3: "have to" or "have got to do something Be obliged or find it necessary to do the specified thing" With the first sentence you are saying that there is some kind of requirement to get up early. (To start work, to catch a plane, etc.) The second is not wrong but would rarely be used any longer. "I am going to get up early" would be more common. It is something that you have decided to do, or perhaps even been requested or directed to do ("Uncle John told me that I am to get up early to accompany him to the train", though a sentence like that would be more at home in the year 1890) but to me it carries less of a sense of obligation than "have to" does.

'I have to' means it is necessary, maybe for a job. You feel like you need to, but it is still your choice. 'I am to' is an old way of saying 'I have been told to' e.g. "The captain said that I'm to head out first thing tomorrow." You are compelled to do something (often by law, or in the military), and have no choice. So the difference is between making yourself get up early and being made to get up early. Most of the time, 'I have to' will be more appropriate. Let me know if this needs clarifying.