Both sentences are perfectly correct. They have a slightly different connotation however.
"Did you go somewhere exciting at the weekend?"
The somewhere in this sentence suggests to me that you are certain that they went out, but you are asking if the place to which they went was exciting.
You might use this form if the person usually went places, or perhaps they are sunburned or look hungover or some other thing that you're sure they didn't just stay at home.
"Did you go anywhere exciting at the weekend?"
The anywhere in this sentence suggests to me that you are uncertain about both whether they went out at all, and whether it was exciting if they did go out.
Note: I am a speaker of British English so there may be other subtle difference to a US speaker of which I am unaware. Additionally, I have seen "at the weekend" used quite a lot in English textbooks for non-English speakers, but I would say that "on the weekend", "over the weekend" or "last weekend" are probably the more common ways to say it for a native speaker.