Already in Hellenistic Astrology, the then known planets were seen as imbued with Aristotelian qualities, which implicitly made them equivalent to the elements. Thus later, for Raymond Lull and subsequently in Medieval Medicine, Mars (being hot and dry) represented Fire, Venus (cold and moist) Water, Jupiter (hot and moist) Air, Saturn (cold and dry) Earth. Mercury is a more complex mixture of qualities; he could be seen as quintessential or etheric, although, according to an influential lore he is predominantly earthy. The Sun and the Moon are representing Fire and Water, respectively, of course.
Thus we get:
I = Quintessence or Earth
II = Water
III = Water
X = Air
XVI = Fire
XIX = Fire
XXI = Earth
There have been various attempts made by modern astrologers to incorporate the trans-Saturnian planets into the aforesaid scheme of the qualities/elements, however, no consensus has been reached so far.
Regarding the outers' correspondences to the Major Arcana, there is no generally agreed upon system either, but the most popular view seems to assign them to those three Trumps which have no planetary associations in the GD system traditionally, so that Uranus now corresponds with The Fool, Neptune with The Hanged Man and Pluto with Judgement (The Aeon) - which quite makes sense astrologically as Uranus stands for unconventionality and unpredictability, Neptune for surrender to a higher force, Pluto for death and resurrection. Also the traditional association of the three Trumps with Air, Water and Fire ties in quite well with this.