Depending on the series, the DVD/BD will have both Japanese and English subtitles, only one of them, or none at all. In addition, certain disks will only show Japanese subtitles if you choose English audio. As such, it'll be best to check upfront whether the disk of the series you're interested in has the subtitles you need.
On the other hand, learning Japanese from any sort of video can be frustrating, especially as a beginner: constantly having to pause and rewind quickly gets annoying. Allow me to recommend visual novels instead: these games always come with subtitles, automatically pause at the end of each line, and have a "Replay current dialogue line" button. In addition, there are third party programs available that'll capture the current line and assist you in reading it (word delimiting, furigana, dictionary lookups). Just be aware that a lot of games in this genre have 18+ scenes at one point or another.
@Mike Cash: Interestingly, it appears the opposite exists as well - Japanese people learning
English through anime subtitles.
日本のアニメで英語学習する方法・コツまとめ – 英語のまんま