I would like to learn what others believe the big bang universe is expanding into? I only want to talk with those who are prepared to explain their own understanding directly, not second parties, and I don't intend to read articles by others that may be posted or linked to on this thread. .Thank you for your understanding.
Before I answer, imagine the Earth with latitude and longitude lines. Now, consider the flow of time to be north on that sphere, so events are at the 'same time' if they are the same longitude.
In this toy model, the south pole acts like the singularity at the Big Bang. Then, as we go north, 'space', consisting of a latitude line, expands until we get to the equator. Then, as we go further north, the latitude lines again contract until we get to a 'Big Crunch' at the north pole.
Now, as I said, this is a toy model, but it is very similar to the simplest actual models of the Big Bang.
Now, imagine you are on a longitude line south of the equator and you ask 'in which direction are the longitude lines expanding?'. The only answer is 'in the direction of increasing latitude', in other words, 'north'.
In the Big Bang model, the universe *throughout both space and time* has a geometry. As we go to later times, the spatial part of that geometry gets larger. This is analogous to how latitude lines get larger as we move away from the south pole going north. Different 'time cross sections' correspond to 'space at those times' and the later cross sections are larger than the earlier ones.
So, the best answer to your question of 'what is the universe expanding into?' is, literally, 'the future'.
One of the biggest difficulties people have with modern physics, especially general relativity (which is relevant for the Big Bang), is that space itself can be curved. The next difficulty is the realization that both space and time join together into a geometry encompassing both of them. To say that the universe is expanding simply means that later 'cross sections' of this geometry are larger than earlier ones.
NB: I am a mathematician and have studied General Relativity and the Big Bang models quite a bit.