I previously asked how to count episodes but I forgot to ask how to say "series". How can I say
I watched 13 episodes of season 1.
You can use 期【き】, シリーズ, 部【ぶ】, or シーズン, all of which are common. Here are the examples taken from Wikipedia (I haven't confirmed, but presumably these reflect the official namings):
These are basically interchangeable, but here's my impression:
You can use these words to count the number of seasons, too, by just removing 第:
And we have another unit to count TV series, クール, which roughly corresponds to 12-13 weeks (episodes), starting from April (春クール, and so forth), July, October and January. It's either from German Kur or French cours, according to Wikipedia. Virtually all Japanese TV dramas and anime are produced under this クール system, and in many cases 1クール exactly corresponds to 1期/シリーズ.
It's unlikely that broadcasting companies officially use クール (it used to be jargon only used by professionals). But now it's widely used among anime/drama fans. For example, you can say: