In addition to the mitchell's answer, it would be much more natural to say "I've seen season one" or "I've seen the first season". I would only add "of the series" if you really wanted to make it clear that you have only seen the first season of the series, as opposed to the movie adaption or something. Otherwise it's implied by context.

I have seen the first (or 1st) season of the series. I have watched the first (or 1st) season of the series. Both look fine to me. However it must be "first" or "1st". Otherwise it would read as "I have seen the one series..."