[verb (dictionary form)] + ようにする [verb ない-form] + ようにする
ようにする means "make a point of", "make sure to": you put effort into doing something, or with the ない-form into not doing it. It follows the dictionary form or the ない-form, and most often appears as ようにしている for a habit you keep up.
ようにする is a decision about from now on. ようにしている is an effort you are already keeping up.
ようにする is the effort. ようになる is a change that has come about, so it is the one that goes with "become able to".
These go beyond your own habits.
As a request, it asks someone to take care to do something, and is softer than a bare てください.
Before a potential verb or an intransitive verb, ようにする is arranging things so that something can happen.