There's actually two parts to your question, which are more or less independent:
"cut the bread" vs. "cut up the bread"
I think this is an area where different people may have different opinions (and it might be region-specific), but to my ear, "cut the bread" sounds better than "cut up the bread", especially when there's more to the sentence (i.e. when you're adding "thin"/"thinly" to it).
Note that "cut up the bread" is actually not grammatically correct (it should be "cut the bread up"), but using "cut up" as a compound verb in this way is actually not uncommon, and I don't think anybody would really consider you wrong for doing it (except some English teachers).
If I could offer a third alternative, I would personally be much more inclined to say "slice" the bread rather than "cut". This is partly because "sliced bread" is a pretty common phrase, and also because "slice" actually has the connotation of cutting off flat pieces ("slices") from a whole, which is what you're usually doing with bread.
"thin" vs "thinly"
Technically, here, "thin" is grammatically incorrect. "thin" is an adjective, and you need an adverb ("thinly") to be correct here.
That having been said, many people would probably use "thin" here, even though it's technically wrong. In terms of naturalness, at least in an informal setting, I think both of them are about the same. My advice, however, is to get in the habit of using the right words as much as possible, because sometimes it can make a difference.
To me, the most natural (and correct) would be:
You have to cut the bread thinly.
You have to slice the bread thinly.
The following are also reasonably natural (though not necessarily as correct):
You have to cut the bread thin.
You have to cut the bread up thinly.
You have to cut up the bread thinly.
For some reason I can't really put my finger on, the following sounds the most unnatural to me. It may just be because it combines two technically wrong grammars so it's arguably the most wrong of all of them:
You have to cut up the bread thin.