So this is a nice question, considering we can figure out a lot of things about other galaxies that seem so unfathomably far away it'd make sense we could figure things out about our own — right?!

Astronomy/astrophysics is an observational science. We do not have the capacity to carry out experiments by nature, of, well, being small humans, meaning we only learn about things by looking at them. What can we learn about our Galaxy looking at it? Turns out, not much.

We know its shape, that's very evident from just the night sky alone. We can observe the black hole at its center, but we don't really know a whole lot about black holes, so all we can say about this one for sure is the one thing we know for all the others: they're old! And with a certain degree of accuracy we can determine its mass, by looking at how other things behave and move around it.

Do we know the galaxy's age? We in fact do! We look at the oldest globular clusters we can find in the milky way, figure out how many heavier elements they have (particularly Be), and from then we can figure out how many stars existed before these and thus — the age.

What about the size and shape, and our position in it? This is difficult... We can look at Cepheids (special types of stars that change brightness allowing us to discern their distance and the distance of objects around them) on the other end of the milky way and determine how far those are, but this really is not accurate. It's like measuring a basketball court by taking the distance between two furthest players. Not very accurate is it? We can't know the way it looks, zero way to do this without stepping outside, and we can somewhat discern where we are...in comparison to other objects in the galaxy? This is also a very rough estimate.

In short, the only real things we can know about the galaxy is the type and age. We can figure out a whole lot about objects in it though. :)