Update #1
Original Post w/ Challenges
So after some deliberation over which types I should pick for my duo-type run, I eventually decided on Water and Flying. I was tempted with Dark, Bug and Fighting but eventually dismissed them.
-I picked Oshawott as my starter and nicknamed him Otto. He has a relaxed nature, which isn't the best but it will have to do.
-I pretty much only used him because if I was only using water and flying types, it meant I had to solo the first gym with him, as I have no access to anybody else of those types.
-I beat N easily in Accumula town.
-After training up to level 15 I decide to test my luck in the gym. I beat the gym trainers easily, bringing me up to level 16.
-After taking down Cilan's Lillipup with little trouble, he sends in Pansage.
-I manage to get his Pansage down to very low health, but unfortunately his vine whips are too strong and I lose.
-I decide to do some more training in route 2 before going back to the gym again. I get to level 17 where Otto evolves into Dewott. Renewed with confidence I go back to the gym, where Lillipup is OHKO'd and Pansage is not difficult to take down in 4 moves (taking into account Cilan's potion)
-ThePhalanx's challenge means I have to teach Work Up to one of my pokemon. Unfortunately, Otto cannot learn it, so I teach it to a Patrat caught on route 1. To keep in spirit of the challenge though, I will teach work up to the first pokemon that will be on my team that can learn it anyway.
-Naix's challenge means I must deposit Dewott until I beat Lenora. As I have no Water or Flying type pokemon, I will use Pansear until I can get another pokemon of those types, which shouldn't be long as Woobat and Pidove come pretty soon.
So I have beaten the first Gym Leader! Which means, I get another challenge.
It would be nice if most of my challenges from now on are one time challenges, that are to be completed by the time I get my next challenge. Please check my original post to make sure your challenges don't clash with them.
EDIT: Challenge for Ziposaki
Ziposaki, your character is obsessed with equality. Therefore for every gym leader you face, each of their pokemon must be KOed by a different pokemon each time. Also, because you want everyone to gain a bit of exp. points, you must be on set mode for all gym battles.