They are very much interchangeable here. The emphasis is on "korán" ("early"), so the exact verb is not that important. "Kelni" is the action of getting up. When the Sun rises, it takes several minutes. "Felkelni" is the total, completed action, at the end of which you are up and running.
In the case of getting up early, it does not really matter, you have to start way too early in either case. And that is the focus here.
"Kelni" (rising or waking) is a general term referring to "getting up" in the morning in this context and it's shorter so Hungarians would use that one more. "Felkelni" would still be correct, but people would wonder why you're getting so specific! :) "Felkelni" is more specific to the act of waking up from sleep. Hope this helps.
"Szeretek" is present tense for "I like" or "I love."
“Szeretnék” stands for “I would like” or "I would love." (Like and love in Hungarian have a very fine line between them.) Would is a modal verb and we often use it, well we can use it for imaginary situations, but in this case, we use it to express a little more politeness. For example, when ordering something in a restaurant, you'd say "Szeretnék egy levest." (I would like a soup.) It is used when you want something specific in that moment or in the future.