So everything was very small, and then it got bigger?
Yes, including space-time.
But it could only be measured on the inside, and if everything was very small, so were the measuring sticks. So it would appear ginormous, from the inside.
Nope. Space-time is defined by gravity. Measuring sticks are defined by other forces, like electro-magnetic forces. Electro-magnetism is 10
37 times stronger than gravity (this is why the entire mass of the Earth is too feeble to break your legs when you are standing on it) so measuring sticks tend to keep their length.
Even gravity itself can stop expansion in local areas if there is enough mass about, for example, galaxies and galaxy clusters.