Short answer: It depends.
When to Keep Category & Tag Pages
If your Category and tags have a lot of great content and match the searcher's intent, it will be helpful and engaging to readers. Then you will want to keep them indexed.
An example of effective use of Category pages (Search Engine Land):
When to NoIndex Category & Tag Pages
It is imperative to avoid these common mistakes:
Duplicate Content: Using similar Categories and Tags in a single post. (e.g., tagging a post with: sell, sellers, selling, vendor, dealer, etc.,)
Link Bloat: Tagging every page with a Category or Tag.
Thin Content: Having Categories or Tags with only one or two posts attached.
If you don't have time to manage your Category and Tag hierarchy properly, noindex might be the best option for you. If you have the time and resources, you might get rewarded for that.
Before Noindexing Category & Tags!
Before you decide to noindex your category and tag pages, check Google Analytics and Google Search Console to see if you receive any organic traffic to those pages.
If you receive little to no traffic over a 6 to 12 month period, it is safe to noindex those pages.