For your completing enjoying any piece of work (not limited to series), you can simply say [verb]-終わる／終える or 全部 [verb]-る. Verbs are different according to medium, so 読む for books, 見る (formally 視聴する) for audiovisual (TV, video...), and 聞く for audio (songs, radio...). ～しまう indeed partially covers such a usage but not easy to use in this situation.
この小説を読み終わった I finished reading this novel.
アニメを13話全部見た I completed watching (all) 13 episodes of an anime.
There are, however, also a few fancier and terser words frequently used: 読了【どくりょう】（する） and 読破【どくは】（する） for completing books, especially the latter carries a nuance that you have accomplished a great goal. Anime fans have a funny figure of speech 完走【かんそう】（する） for watching the whole season, which is otherwise used when finishing a marathon run or such.
For the second question "a series is completed", I think most people solely choose the verb 完結【かんけつ】する "(finish and) conclude" to describe any serial story to reach an end.