Technically I had two: The N64 and the Sega Genesis.
I myself owned the Genesis when I was 3 or 4, and it was the original model. I can't remember, nor can my parents remember, where they got it for me, but it was the first console that was constantly in my bedroom for me to play whenever I wanted.
The N64 was my father's, being brand new and more expensive and all that. This was where I was introduced to my favorite FPS series of all time, the Turok games. However I could only really play it when my mom left for work and my Dad was around to make sure I didn't break it, though somehow at some point my Dad gave up altogether and just left me to use it alone, probably because the system didn't really have any moving parts to break.
The first console I bought with my very own money was a PS1, when the PS2 was already out for about a year or two. It was only $50, so I didn't need to save too much.
And like 98% of everyone else, my first handheld console was a Gameboy, and then a Gameboy Color.