Great question, not so easy to answer.
Without internet access in the 1980s and early 1990s, you only read about a new game 1 - 3 months after the release, if there wasn't a preview article.
I'm pretty sure, that I bought the C64 version of "Dragon Wars" instantly after I read the game review (because I loved the predecessors "The Bard's Tale 1 - 3" so much)... but the earliest reviews in German game magazines were in the February issues of 1990, the game launched in November 1989.
The game had one of the coolest box covers ever:
Inside the box there was even a poster of the Boris Vallejo cover art, which was hanging on the wall of my room a few years.
It was also one of the very rare C64 games with a C128 mode, which (slightly) enhanced the game.
----------------------------------------------------------------------------------------------------------------------------------------
The first game I bought "day one" of its release was the German Amiga version of "Monkey Island 2: LeChuck's Revenge".
I preordered it in a software shop two towns away and on the day it arrived (June 5th 1992) they called me. Of course I collected it the same day, I still have the receipt in the big box.
A few years ago I met the adventure legend Ron Gilbert and he signed my Monkey Island 2 box and the code wheels of MI1 + MI2.
The German translator of the early Lucasfilm/LucasArts games (Boris Schneider) was also there and he signed the MI2 box, too.
But even if I bought this version of the game at its launch day... the English Amiga version was already available a few months earlier and the English MS-DOS-version even since December 1991... so half a year before I bought the German version.
----------------------------------------------------------------------------------------------------------------------------------------
So technically, the first game I purchased at launch was the CD-ROM "Full Throttle" (German name of that LucasArts adventure is "Vollgas"), which I bought in May 1995. That version already had the German box cover, handbook and strategy guide, but on the disc was still the English version of the game.
The publisher Softgold offered an upgrade to the localized game for 30 DM (~15 Euro) so the second big box with the new disc (German interface, German voices and German subtitles) was sent to me in September 1995.
----------------------------------------------------------------------------------------------------------------------------------------
The first console games I bought day one (of the EU release) were "Donkey Kong 64" (December 2000) and "Conker's Bad Fur Day" (April 2001).
Unfortunately I sold them a few years later. Especially Conker is now worth hundreds of Euros, even a used cartridge.
----------------------------------------------------------------------------------------------------------------------------------------
The first handheld games I bought together with the hardware at system launch were the PSP games "Ridge Racer", "Tiger Woods", "Untold Legends" and "Gretzky NHL" (the pack-in game). I imported the PSP-1000 and those games from Canada, the packages arrived in April 2004... so half a year before the EU-launch of the PSP! I was lucky, and didn't have to pay any customs duties.
----------------------------------------------------------------------------------------------------------------------------------------
The first home console games I bought together with the hardware at system launch were the Xbox 360 games "Perfect Dark Zero" (Limited Edition), "Kameo: Elements of Power", "Project Gotham Racing 3", "Condemned: Criminal Origins" and "Peter Jackson's King Kong".
The Xbox 360 almost had a global launch (EU launch only a week after the Americas and one wekk before the Japan launch) and I was very hyped. The only console lauch where I preordered the hardware and launch games weeks ahead and stood in line on the launch day (December 2nd 2005).
