if i may a few fixes:

at the end change the line

http.ListenAndServe(":3000", router)

with

http.ListenAndServe(":3000", handler)

and the api as by convention it should be more like:

router.HandleFunc("/healthz", Healthz).Methods("GET")

router.HandleFunc("/todos/completed", GetCompletedItems).Methods("GET")

router.HandleFunc("/todos/incomplete", GetIncompleteItems).Methods("GET")

router.HandleFunc("/todos", CreateItem).Methods("POST")

router.HandleFunc("/todos/{id}", UpdateItem).Methods("PUT")

router.HandleFunc("/todos/{id}", DeleteItem).Methods("DELETE")

this is a very good article on how to REST API should be written