In chinese the basic syntactic structure is:
主语, Subject + (状语, Adverb) + 动词, Verb + (补语, Complements)
So, from this structure we can see that the adverb comes before the verb and the complement slot follows the verb.
Whatever precedes the verb modifies its meaning. So, since the adverb comes before the predicate, it behaves like a determinant. 很快(地) in your first sentence adds more information to the main action, 看.
On the other hand, whatever follows the verb represents the results of the predicate, it completes the meaning of the verb and that's why this place is reserved for complements. The degree complement, or 程度补语, it's a very special complement that specifies the qualities of the action. You could think about it as picture of the predicate.
Thus, both your sentences are correct from a grammatical point of view. They also mean the exactly same thing. The difference between the two consist in the focus.
In the first sentence the predicate (看) is the most important element. In the second instead the degree of the action (得很快) receives the focus.
In general I read messages. How do I read them? Quickly.
In general I read messages and my action has the result to be quick.