I started with the Master System in '87. It came with Hang On/Safari Hunt, and the first other games I had were The Ninja and Ghostbusters. I still really enjoy playing The Ninja from time to time as it's quite a cool game which stands up today, in addition to being like an intense blast of nostalgia for me.
I remember sitting there with Christmas catalogs, making the decision between the Master System, the NES, or the Atari 7800. Even though our family's first console was the 2600 and I had a soft spot for Atari, I could easily tell it didn't match up to what Sega and Nintendo were bringing out. Ultimately I went with Sega because of their arcade pedigree. I had known them from my own early days in the arcade with Pengo and Zaxxon, and the badass Subroc 3D... but of course, more recently with Hang On, Out Run and After Burner. I don't think I knew too much about business then, but I was savvy enough to deduce that those games would only be coming home from Sega. I think the next year is when they came out, and I got them all, as well as Space Harrier. That was my first experience of Space Harrier, for the SMS... it wasn't until a year or two later that I finally found one in an arcade.
Other Master System games that I eventually got were the cool and creepy Kenseiden; Miracle Warriors, my first console RPG; Aztec Adventure; Rambo, a great riff on the Ikari Warriors formula; Rocky, with its amazing huge character sprites; Shinobi; Alien Syndrome; Ghost House; Double Dragon (my NES friends practically shat their pants when they saw that we got a 2-player version while they were stuck with their nonsensical single player Double Dragon!); Quartet, one of the first games I beat; and R-Type, a very difficult shooter that compelled me to keep getting further just because the artwork was so amazing.. I was also fortunate to have a couple of friends who also had Master Systems, and could borrow awesome games like Golvellius, Rastan, Time Soldiers, Thunder Blade, Alex Kidd: the Lost Stars, Maze Hunter 3D with the 3D glasses, and of course Phantasy Star. I don't care what anyone says, it had a solid library of excellent games. I would occasionally trade systems with a friend with an NES, and never regretted having chosen the Master System.
I think each generation of consoles has its own 'magic' if you will, but yeah, the Master System days were amazing. Having been through that time made the success of the Genesis all the more sweet, too.