よみます = to read (infinitive: よむ)
かのじょはひるごはんのあとによみます is correct but it sounds unnatural to me because よむ needs object like "read the book / newspaper / etc". I would say: かのじょはひるごはんのあと、ほんをよみます。
*We have another way of saying "to read the book(s)", it's どくしょをする and どくしょ is a noun for "reading book(s)", so you can say "I like reading": わたしはどくしょがすきです。
のみます = to drink (infinitive: のむ)
かのじょはひるごはんのあとにのみます can be understood "she drinks (alcohols) after the lunch", in other cases, we need object like "coffee / tea / juice / etc".
きます = to come (infinitive: くる)
くる is the verb which conjugates irregularly.
I hope this was helpful but if you have doubt, please let me know.