There's a lot of good Japanese content on all the major streaming sites, e.g. Netflix, Hulu, Amazon Prime, etc.
They're not free, of course, but it's all usually very reasonably priced (usually around $10/1000 yen a month), and given the wealth of content and the quality/convenience of the video, I'd say it's very, very worth it.
The Japanese words for (television) drama and anime are, indeed, ドラマ and アニメ, respectively.
The Japanese language technically doesn't have singular and plural forms (at least, the concept as you think of it in English and other languages doesn't really apply), so the words ドラマ and アニメ could refer to a single series, multiple series, or just anime and tv series in general.