13.7 billion years ago the singularity appears and begins expanding at the speed of light. Am I right so far?
After a while … a long while … things cool down enough for matter to form. Some of that matter makes up the known/observable universe. (Let’s not complicate things with dark matter for now.)
Where is the universe? On the surface of the expanding singularity? Inside the singularity?
Grab a point (any point, it doesn’t really make a difference), and draw a sphere centered there for the 13.7 billion years * speed of light. That’s the observable universe. (for some value of ‘observable’)
It’s best to thing of the Big Bang Theory as backward-looking. Under the model, space contracts as you look back in time, becoming hotter and denser. The model holds until a fraction of a second from the time when distances shrink to zero and density and temperature go to infinity. But this “singularity” is just the point when the model breaks down, it is not something real, it is not a prediction of the model.
Prior to Big Bang expansion, we don’t know exactly what happened. There was almost certainly a brief moment of more rapid expansion called inflation, and we may have definitive evidence of this in the near future. As to how this universe, or any universe, gets started? There are some interesting and highly speculative ideas, but we may never know.
It’s important to be clear the the universe is not something appearing and expanding in pre-existing space. The universe is the space, and it is the space itself that is expanding. Space is not expanding at the speed of light - it is not expanding at any one speed. Points farther away in space are receding from one another more rapidly than points closer together. Points that are far enough apart are receding from one another faster than the speed of light.
It’s misleading altogether to think of the Big Bang as a single point “in” space. If you look back in time, everything shrinks. But that doesn’t mean things are shrinking within some “external” space. The space itself is shrinking. Counterintuitively, this means the Big Bang happened everywhere.
The observable universe has a radius of about 46 billion light years. The expansion means that points in space that are now 46 billion light years apart were closer together in the past.
The singularity is a description of the universe as it appears to have been 13.7 billion years ago. The relationship between the singularity and the universe is one of time, not position.
Let’s say you could magically stick a flag at some point in space, and that flag would never ever move. Now you stick another magical flag at some other point in the universe, and that flag also never moves. What you find is that the two flags are moving apart from each other, even though neither flag is moving. What’s actually happening is that the space between the two flags (and the space everywhere for that matter) is expanding.
If you go back in time and rewind this, the space itself contracts all the way down to a point. That point where you stuck the first flag ends up in that point. That second flag also ends up at that same point. A magical flag in New York City ends up at that point, Some random flag in the vicinity of Alpha Centauri ends up at that point. Every point in the universe all ends up at that same point, because every point in the universe gets closer and closer until it all squishes down to that single point, aka the singularity. Not just the matter, every actual point in space itself (all of those imaginary unmoving flags) compresses down to that single point.
Asking where is the universe is kinda asking where the universe is in space, but space itself also started at that singularity, so in that sense the question itself doesn’t work. Instead of where is the universe, the question is more like what is the universe. Is this expanding spacetime actually in something else, something that we can’t really comprehend because we have only ever experienced our own spacetime and our universe? Are there multiple universes out there? (Stephen Hawkings seems to think so)
Currently, about all we can say for certain is that we don’t have a freaking clue about what is beyond the universe, and what the nature of the universe’s existence is.
You’re soaking in it!
Ha! Brilliant.
Let’s say the universe is “here” … and there is no “there” … such that “here” means everything.
The Universe is the mystery that had no beginning and has no ending. 
Words like “here,” “there,” or “where” refer to spatial locations, i.e. locations within the universe. To refer to “where” the universe itself is, is meaningless. It’s like asking “When is time?”
Right here. Right now.
It isn’t mystical, it’s just the simple fact that any coordinate system we can define must be defined in relationship to the Universe, and it loses meaning outside of the Universe.
Asking where the Universe is is like asking the latitude and longitude of the Earth itself: Latitude and longitude are used to pinpoint places on the Earth, so literally every point you can describe using that coordinate system must be on the Earth. Similarly, asking how to describe times before or after the Universe would be like asking what’s north of the north pole or south of the south pole: Starting at the Big Bang, literally every direction is “away from the Big Bang”; that is, into the future. Black holes work the same way, in that once you cross the event horizon, every direction you can travel is “towards the singularity” because the immense gravity has warped space-time such that escaping the event horizon is the same as going back in time.
No other place I’d wanna be.
Also, does anyone know how soon is now?
By the time you ask, it’s ago.
I searched for the outside of the universe, and when I got there, there was no there there.
I’ve heard that human brains are biologically programmed to produce the answer to that question just before they die.
“I’m astounded by people who want to `know’ the universe when it’s hard enough to find your way around Chinatown.”
I understand your question, and I’ve been asking myself analogous questions. Why is the Universe right here, instead of somewhere else? Why did the big bang happen when it did, and not earlier or later?
It’s everywhere, so the question is moot.
Also somewhat moot, given that effectively time itself started with the Big Bang. And if that idea blows your mind you’re in very good company.
There are certainly various theories that include an “instigating event” of sorts (branes theory does IIRC) but given the “time started with the Big Bang” thing it involves a complicated understanding of “instigation”.