(Warning: Wall of Text approaching)
If you count consoles, my first gaming system was the Atari VCS (aka Atari 2600 in later years). There are some timeless classics on that system that I still enjoy playing today, especially Pitfall and River Raid. Pretty much anything from Activision was a safe bet. My dad bought us the original "Heavy Sixer" model, complete with faux wood grain trim. My brother and I played the heck out of that system and dad eventually had to replace it with the later "Darth Vader" model (named for its all black appearance and slight resemblance to the Dark Lord of the Sith). My dad knew a guy at work with a ROM duplicator, and every Friday night he'd come home with a case of chips that we'd plug into a special cartridge that played duplicated games. We'd still buy games we enjoyed, though. It's funny, back then we had no idea about things like DRM or copyright, but we still knew that if you liked a game you should buy it to support the developers, and if you didn't like a game enough to want to buy it then you should stop playing it. We didn't need some draconian government organization to tell us to "do the right thing."
The first computer I played games on was the Coleco ADAM. It was essentially a hopped up Colecovision console with added memory, tape storage, keyboard and printer. It also had a cartridge port and two built-in joysticks, so it could play all Colecovision games. Much like the Commodore 64, it was one of the console/computer hybrids of the early 80s. Unfortunately, the thing suffered from some serious design problems. The biggest issue was that whenever you'd turn on the thing, it would emit a massive electromagnetic surge which over time would start to erase any data cassettes you left in the tap drives or even next to the computer. It didn't help that some programs advised you to put the data cassette in the drive then turn on the computer to boot the program. ADAMs also had a pretty high failure rate. I remember taking ours in to a service center at least two or three times because various components blew out. But the games were awesome -- even better than a Colecovision and easily on-par with a Commodore 64. The ADAM didn't have anywhere near as big of a catalog as the C64, though, but the games it did have were all very high quality for the time.
The first "PC compatible" system I used for gaming was an Intel 80286 based system with 640 KB RAM, 5.25" 1.2 MB floppy drive, 30 MB internal hard drive and CGA graphics (640x480 with a whopping 4 colors!). Of course, I didn't buy it specifically for gaming. My high school offered a series of programming courses (BASIC, Pascal, C, FORTRAN and COBOL), and they all used PCs. So my parents very graciously bought me a mid-range PC clone system. The games definitely paled in comparison to other home computers of the time, and especially game consoles. Aside from the 4 color graphics, PCs didn't have sound cards yet so all you got were primitive beeps and boops from the internal speaker. I didn't really do much gaming on PCs until the advent of Super VGA graphics and the Sound Blaster audio card.
The first system I built specifically for gaming (namely, to play TIE Fighter and Warcraft 2) was an Intel 486DX2-66, with 8 MB RAM and some variety of Diamond Stealth graphics card (one of their high-end ones from the early 90s). I paid a lot for that system, but it ran every game I threw at it for several years smooth as silk. The first games I played on this system included X-Wing, TIE Fighter, Warcraft 1 & 2, Doom and Wing Commander. Some of those I also played on my older 386SX system first, but that system didn't have a very game-worthy graphics card so most of the games really chugged along. That's what prompted the upgrade to the 486DX2, and then I was in gaming nirvana until a little gem called MechWarrior 2 Titanium Edition convinced me that I needed a Pentium CPU with a Voodoo graphics card.
Ah, good times, good times.