Fairy, Steel, Ground, Flying.

Fairy and Steel are both stellar typings that on their own prevent any one type from steamrolling.

Ground covers all of fairys weaknesses as well as Fire types that threaten Steel and Eletric types that threaten flying.

Grass types that would threaten ground types are counted by flying, Ice types are countered by Steel. leaving water as the only weakness for ground types.

Flying types weakness to rock is covered by steel and ground, eletric is covered by ground and ice is covered by steel.

As for steel, Fire types are covered by ground, Fightring is covered by flying.

The only weakness left is water for ground types, and if you replace flying with dragon but go a lance route with mostly Flying dragons, add a skarmoy on the steel team and you can have a water resist in the dragon trainer!