Please note: The following is only my guesses according to observation. I don't claim to be right. I just claim to have no life.
According to tradition, the starters will most likely still be Grass, Water, and Fire, and the first Gym will be easy for two of those types and somewhat difficult for the third. (Brock/Roxanne's Gym was easy for Squirtle/Mudkip and Bulbasaur/Treecko but harder for Charmander/Torchic, and Falkner's Gym was easy for Totodile and Cyndaquil but harder for Chikorita.) With that in mind, it'll probably still be something easy for two of the starters but difficult for the third.
Hence, I predict that the first Gym will be Electric. Electric can be beaten without a problem by Grass-types (which have a resistance to the type) and Fire-types (which is neutral to such Pokemon unless an Electric/Steel Pokemon like Magnemite is involved, at which point there will be an offensive advantage in Fire's favor), but Water-types, while having no offensive advantage or disadvantage against Electric, is weak to Electric-type moves (thus making the Gym difficult).
Alternatively, the first Gym could be Ground, which is pretty much the same, only switch Water and Fire in the above paragraph. This might be the better choice, seeing as the Electric type already has two Gyms. (See below.)
Beyond that, I'd have to decide by looking at what types of Gyms already exist.
Normal: Two Gyms (Goldenrod and Petalburg)
Fire: Two Gyms (Cinnabar and Lavaridge)
Water: Two Gyms (Cerulean and Sootopolis)
Electric: Two Gyms (Vermilion and Mauville)
Grass: One Gym (Celadon)
Ice: One Gym (Mahogany)
Fighting: Two Gyms (Cianwood and Dewford)
Poison: One Gym (Fuchsia)
Ground: One Gym (Viridian)
Flying: Two Gyms (Violet and Fortree)
Psychic: Two Gyms (Saffron and Mossdeep)
Bug: One Gym (Azalea)
Rock: Two Gyms (Pewter and Rustboro)
Ghost: One Gym (Ecruteak)
Dragon: One Gym (Blackthorn)
Dark: No Gyms
Steel: One Gym (Olivine)
With all that in mind, the games will probably go for types that either haven't been covered yet or have only been covered once, just to avoid redundancy (although a Water Gym seems to be a bit of a tradition). Thus, the following types I predict will be seen, in no particular order, are the following:
Ground
Dark
Steel
Grass (or Poison)
Ice
Bug
Ghost
Dragon
With a possibility of a Water-typed Gym to continue tradition.
Now, for the fun part. Putting them in order.
I've already stated my reasons for thinking Ground will go first. The type after that usually displays more of an advantage with one type over another. To be specific, the Gym after Pewter was Water, meaning that though Squirtle and Bulbasaur had an advantage over Brock (considering the RBY series, in which Metal Claw isn't available), Bulbasaur had more of an advantage in Misty's Gym than Squirtle. Meanwhile, while Cyndaquil and Totodile did better than Chikorita in Falkner's Gym, against Bugsy, Cyndaquil had more of an advantage. The only League that doesn't fit this pattern is Hoenn, in which a Rock-type Gym follows a Fighting-type Gym, meaning both Mudkip and Treecko had equal footing there.
Even so, considering the above, I predict that the Gym following the Ground-type Gym will be a Bug-typed one, as a Water-type Pokemon would have more of a footing there than a Grass-type would.
After that, the next Gyms can pretty much fall in any order, but the seemingly most difficult Gym would be placed last (obviously). It was true with Clair, it was true with Giovanni (although that was mainly because of his Pokemon), and it was true with Wallace (although that was mainly because of Milotic). Nevertheless, I've put Steel down as the final Gym, with Dragon being the seventh.
Thus, the final list is as follows:
1. Ground
2. Bug
3. Grass (or Poison)
4. Ice
5. Ghost
6. Dark
7. Dragon
8. Steel
