When was the Milky Way galaxy discovered?
It wasn't until telescopes that people realized that the band of light reaching across the sky, called the Milky Way since ancient times, was actually made of an immense number of stars. Astronomers still did not really understand what they were seeing until the 20th century, however.
Until the 1920s, astronomers thought that what we now know to be our Milky Way Galaxy to be the entire universe, and that our whole universe was a few thousand light years across. Other "spiral nebulae" had been observed, but they were thought to be new star systems forming nearby. After Hubble (the astronomer, not the telescope named for him) observed Cepheid variable stars in the Great Nebula in Andromeda, he realized that the Andromeda "Nebula" was immensley distant, and ennormous in size, and, by extension, the other "spiral nebulae" were also huge and incomprehensibly distant. He called them "island universes", and realized that we were also in one, and that the 'Milky Way' band of stars across the sky was our galaxy's disk, seen from inside. So, even though people have been calling the band of light across the sky the Milky Way for thousands of years, it wasn't until the 1920's that we understood what it was--our galaxy!
We can see only a small part of our galaxy in visible light. Since the 1960s, radio astronomers have mapped out the structure of the entire galaxy, and shown it to be a large spiral galaxy of about 100 billion stars; we are in one of the spiral arms about 8 kiloparsecs (25,000 light years) from the center of our galaxy, more or less halfway from the center to the edge.
I think Gallileo came across the milky way in the 1600's. The cloudy band we now call the Milky Way has been known since ancient times (it's referenced in various cultural mythologies, for example). However, it was only in the past few centuries that it was properly identified as a galaxy, specifically our own.
== == The Milky Way's true age hasn't been discovered. The only
knowledge we have is of a meteorite which dates 4.7 billion years
ago. And yes, Galileo discovered the odd colors of the Milky Way in
the 1600's. If we could escape our galaxy, scientists believe it
would look like M-31(The Great Galaxy of Andromeda)
I would hesitate to talk about a "discovery" of something that is in plain sight - that people have been seeing for ... well, for as long as there have been people. Like the Sun, or the Moon, or trees, or animals - or the Milky Way.