I'm thinking of joining the Army as a 27D. After completing AIT, am I then made a Specialist with the bump in pay?

Private (E-2)—Six months time-in-grade (TIG) as a private (E-1).
Private First Class (E-3)—Four months TIG as a Private (E-2) and 12 months time-in-service (TIS).
Specialist/Corporal (E-4)—Six months TIG with 24 months TIS

Those are the time in grade and time in service requirements for your first 3 promotions in the Army. If the unit is undermanned in specific grades, the Army may allow the unit commander to waiver TIG and TIS requirements. When specifically authorized, the commander can waive up to 2 months TIG for promotions to E-2, 6 months TIS/2 months TIG for promotions to E-3, and 6 months TIS/3 months TIG for promotion to E-4.

You can come into the service at a higher pay grade for things like being an eagle scout or having a college degree.

The "Specialist" part of the MOS title doesn't mean you automatically get bumped up to the Specialist rank just for completing AIT.

No. You will have to meet Time in Service and Time in Grade for SPC before you get advanced to SPC. Unless you already have a college degree and enter the service as a SPC already, that is.

I’m an active 27D. If you’re good at your job, it will be easy for you to compete for one of the waivers previously mentioned. Our corps is under strength as well, so as a result promotions to NCO grades have increased significantly.