"During" is a preposition. "During the weekends" is a prepositional phrase being used as an adverb to modify "I am too lazy to do anything but sleep."
English permits adverbial modifiers to be placed in a variety of positions. Both of your versions are correct and natural.
The difference is emphasis. Readers will naturally attend to the information that appears toward the end of a sentence. So the version ending with "sleep" emphasizes "sleep." The version ending with "weekends" emphasizes "weekends." You should choose the version that matches the point you want to emphasize.
Another consideration is length. The following is technically correct but a little unnatural.
I am too lazy to do anything but sleep on my new orthopedic mattress during the weekends.
The following sounds a lot better:
During the weekends, I am too lazy to do anything but sleep on my new orthopedic mattress.