Short Answer: There is no straight forward yes/no answer to this question. If you are running a typical Blog using WordPress, where your category and tag pages only contain excerpt from belonging posts, then the answer is no, you shouldn't allow search engines to index your category and tag pages. If you have unique content to those category and tag pages, then the answer is yes, you should index them for better SEO.
Noindex Category & Tag pages if they contain only post excerpt & no unique content
Here is why you shouldn't index tag/category pages with no unique content & only excerpts from posts: