We had a Spectrum (the +2 model, after Amstrad took over) and dozens of games floating around when I was a kid... so undoubtedly my first game would've been one of those.
The first proper games console I ever got was the Sega Master System, the one with Hang On and Safari Hunt built in. I got it just in time for the release of the original Sonic The Hedgehog.
Incidentally, some time in the mid-to-late 90s, I was working in a charity shop when someone donated five assorted Spectrum machines and over 300 games. Because charity shops can't sell untested electrical equipment, they were destined for the bin until I rescued them. Sad thing is, I haven't a clue where they are now... though I have a suspicion that they got thrown out after I left for university.