[verb (dictionary form)] + ことができる
ことができる means "can" or "be able to". It follows a verb in the dictionary form, and できる at the end takes the tense and the politeness.
It also covers what is possible or allowed somewhere, not only what a person is able to do.
The potential form (はなせる話せる, よめる読める) says the same thing in one word. In conversation it is the usual choice; ことができる sounds more formal and turns up in writing, announcements and explanations.
The two do not combine. The verb before こと is the plain dictionary form, never the potential.
The object keeps を with ことができる, since できる attaches to こと rather than to the verb. Only the potential form lets を change to が.
These go beyond a verb before ことができる.
A noun takes できる directly, with が and no こと: にほんご日本語ができる means "can speak Japanese", てにすテニスができる "can play tennis".
For a する-verb, the potential form is the noun plus できる, so 運転することができる and うんてん運転できる say the same thing; the second is shorter and more everyday.