I'm trying to get the requirements to get into the Blue Building, but I can't for the life of me figure out what it means by having a pet with an overall rating of 100 or more.

Could someone please tell me what this means?

Overall pet rating is when you add your pets Level, Strength, defense, and health together, if it sums up to 100 or more, that requirement is met

Speed and intelligence don't count then?

intelligence isn't counted. I am not sure about the speed as it is a new feature

No, those stats won't be counted. I tried it myself and no chance ^^

I turned my pet awsome with an elixir and a cool minion and got the achievement. good luck to you.