The first digital games I bought were the first 11 "Sam & Max" episodes directly in the Telltale Store. I bought the first season at November 1st 2006 after reading the reviews for Episode 101 (and before the release of the other episodes). Beginning October 2007, I always preordered the freshly announced seasons of point&click episodes due to the quality of the games I already played... so I was able download and play through each episode right after its release. 

The first two episodes I still had to download with a slow 56 kbit/s modem over a 64 kbit/s-ISDN telephone connection. Fortunately we got DSL in the first week of 2007... Hallelujah! 

The Telltale Store was very customer friendly back in those days. The downloads were DRM-free, they updated some episode downloads after I contacted them about an audio compression problem and they offered physical versions of the seasons (as soon as they were completed) for just the shipping costs!

Of course I paid the shipping costs for such a great deal:

The disc versions had a disc check, so when starting an installed episode, the correct game disc had to be in the drive.

The discs of Sam & Max Season 1 + 2 are hybrid DVDs: if you put them into a DVD player, you could watch a lot of video material (f.e. cutscenes with designer + production commentaries, shorts, trailers, bloopers and a Christmas Special). Wallace & Gromit and Tales of Monkey Island have similar videos in the WMV format on the disc, they don't support DVD or Blu-ray players. All discs have a "Goodies" folder with concept arts and other stuff.

The Telltale Store is still online 20 years later and my DRM-free episodes are all still available to download:

I activated my Steam account the day the PC Big Box of "Half-Life 2" arrived at my home (November 25th 2004).

But the first digital game I bought on Steam was the rhythm game "Audio Surf" at November 5th 2008... already on sale (-72%) nine months after release.

The first digital game I bought on Steam "Day One" was the MI1-Remake "The Secret of Monkey Island: Special Edition", after that the collection "Indie Games big bundle" and the awesome "Trine" (which looked gorgeous in stereoscopic mode with my Nvidia shutter glasses): 

Since the Christmas sale 2009 I am addicted to Steam.

I recall downloading some Devil May Cry 4 background wallpaper from the Playstation Store back during 2008. On my CRT monitor, the only notable difference between the HD and SD variants of this wallpaper, was that the HD wallpaper was "widescreen"

The first digital game I downloaded was Final Fanatasy VII on the Playstation store during 2012.

I think it's Fallout new vegas on Steam like 10 years ago

I don't remember the exact games, but it would have been Xbox Live Arcade games on the Xbox 360.

Don't remember buying an XBLA game before I got PS3... So might have been something like Blast Factor or Stardust HD on PS3.

I got defend your castle on Wii also.

My first digital purchase was, I think, Zuma not long after getting the 360 at launch lol. Such things were a novelty at the time, at least in the console world, so I ignorantly welcomed the "future". If only I had known...

Yoshi’s Story on the Wii Virtual Console. [2009]

Some backstory: I come from a big family (youngest of all my siblings), and some of my siblings were growing up around the time of N64/PS1/SNES. That said, we owned a lot of N64 games and my siblings had an N64… but they never let me play it, as the time I had reached my “gaming age,” they had already moved out for college.

That said, there were always N64 games lying around the house, but no system to play them with. The art on the Yoshi’s Story cartridge always captured my little 3-4yo eyes. One day as I was screwing around on my Wii, I found the game hidden away on the WiiWare shop. I asked my mom and dad if I could purchase, they agreed, and thus my dream of finally playing this mysterious game had been fulfilled.

I got stuck on the first level and never picked it up again. I was so confused by the gameplay lol.

The Legend of Dragoon for the PS3 on the soon to not be named PSN, back in 2014.

Lylat Wars on Wii's Virtual Console in May 2007

First new game was Lost Winds on WiiWare in May 2008

First Steam 'purchase' was Portal, it was free at the time, shows as May 2010 (I thought it was a year or two before that but that's the date it shows on Steam)

Not sure why they were all in May.

First payment on Steam was December 2010 for Deus Ex + DE:Invisible War, never actually completed the first one I think it was a little too dated so without me having the nostalgia of playing it before I wasn't enjoying it, so never even started DE:IW.