If you are running a typical Blog using WordPress, then the answer is no, you shouldn't allow search engines to index your category and tag pages.
Here is why:
Search engines rank your pages using internal and external linking to and from your pages: higher the incoming link, higher the rank + higher the rank of the incoming links higher the rank of your link.
Each search engine has it's own ranking algorithm, but the basic idea is the same. You may not have any control over the external linking, so I'm giving an example only using internal linking. say you have three blog entries:
http://YourSite.com/wordpress/, http://YourSite.com/Yahoo/ and http://YourSite.com/Google/
and these blog entries are under the category named sites.
Now notice that, the sites category page will have linking from all three blog posts, but each of these posts will be linked only once from the category page itself. So, search engines will give the category page higher priority!
So in essence, if users search something and find your site in the result, in most cases they will find your category page instead of the post that contains the original information. This is a typical case. Because in a typical installation, Category pages of your blog contain only excerpts (i.e. a short description) from the original posts.
So, for WordPress Category SEO, it's safe to index your blog entries only & make category pages noindex
The same applies for WordPress Tags SEO as well. Because in a typical installation, Tag pages of your blog also contain only excerpts from the blog posts. So you almost always gain nothing by indexing your tag pages that only include excerpt.
So make the tag pages noindex as well, for better blog tags SEO.
Same applies for blogs other than WordPress as well. However, if you run a different type of site (something other than blogging), then this decision will solely depend on the type of your site.
In fact it depends on this question:
Do your category or tag pages have additional content other than the posts or do they have added value to your site in any other meaningful way?
If the answer is yes, then go ahead and index them. If the answer is no, then don't index your category and tag pages.
There is a very good WordPress plugin named Yoast SEO that can manage all these for you. So install this plugin and configure everything you need from the admin panel of the plugin.
Remember that search engines will still crawl your category and tag pages, regardless of your indexing in the XML sitemap or search bot setting (either in meta tag or in robot.txt file). The only effect of not indexing them is that, they (category and tag pages) will not have higher ranking over the individual blog entries and pages.
Also, note that while giving the above suggestion, I've assumed that in your WordPress installation (or any other blog), the category and tag pages have a higher degree of incoming internal links. This can differ from theme to theme, but in most cases they do have higher incoming links than individual entries. So, in most cases, it's better for your site's SEO if you index only the blog entries and page entries like about, home, contact us etc.
You are all set, now make use of this WordPress blog tags SEO & blog category SEO tips. Happy blogging