Other than the two main types of God Slayer Magic seen in the Anime and Manga series' (e.g. air and fire), what other elemental magics are there?
Fire, Sky and Lightning god slaying Magic.
these are the details of the episodes where I saw them: season 4 - ep 5 is where you can find the fire god slayer.
Season 6(episode 169) is where you can find Sky God Slayer.
season 7(episode 185) you can find Lightning God Slayer Magic(Orga). I don't follow Manga much so this is how i came to know about them
From what I can tell (solely going by the Wikia), there are three main kinds of God Slayer Magic; Air and fire that you've already mentioned and Lightning God Slayer Magic seen in the manga in episodes #273 and #320.