Well, a gyarados with bite and dragon pulse have resist to vaporeon and arcanine (but no advantage) and have advantage over dragonite. Its an all round against the 3. But if you want a team, there are lots of possibilities, depends os how common are the pokemons too...
For low level dragonites you can have cloyster with the ice moves or clefable with fairy types and for the high level ones, lapras.
Arcanines - Vaporeon is the best choice, can be blastoise, poliwrath, golduck, hydro pump gyarados and any other strong water
For vaporeon the best is grass over electric, nice choices are victrebel, exeggcutor, venasaur and vileplume
Low level gym:
-Easy to find vileplume against vaporeon
-Easy to find clefable against dragonite
-Easy to find vaporeon against arcanine
High level gym:
-Lapras for dragonite
-Vaporeon for arcanine
-Venasaur or victrebell for vaporeon
Im assuming that you want just pokes with type advantages... Beside this, you have all round strong pokemons like Snorlax, exeggcutor and slowbro that can handle a good fight against any poke that dont have advantage over them....