I have many of these. Here they are in chronological order, as best as I can remember.
1.) Renting a Japanese DC and Sonic Adventure months before release. The graphics were amazing, and the game was too. This had me really excited for the DC's release.
2.) Going with my father to pick up the Dreamcast at Midnight 9.9.99, my first and only "midnight launch." I have since vowed to not attend any other midnight launches unless Sega reenters the hardware market.
3.) Finishing Sonic Adventure and playing as Super Sonic. I also enjoyed countless hours of Soul Calibur and Hydro Thunder. The US DC launch title lineup never ceases to amaze me.
4.) Signing up with AT&T to get my DC online and free keyboard. I think I cancelled after a month or so, as there wasn't much to do online with the DC only a month after release. That keyboard would come in handy later.
5.) Waiting forever for Rainbow Six to come out. I had this on pre-order for a while and I just remember it took forever to finally get released. I was disappointed with the game too. It was over my head; I blame part of that on the hard control scheme. This isn't necessarily a fond memory, but one that definitely stands out.
6.) ODCM (Official Dreamcast Magazine) - This magazine I thoroughly enjoyed, and probably the only one I would ever read cover to cover. Not only was the content in the magazine good, but the included demo discs were great as well. After they closed up, my subscription transfered to Next Gen. After Next Gen closed up, my subcription transferred to PSM. PSM was terrible.
7.) Snow days playing Dreamcast in the basement. Worms Armageddon and Tony Hawk's Pro Skater stand out as titles I would play. Tony Hawk was just amazing when it came out - especially on the Dreamcast.
8.) Phantasy Star Online. If you were around to experience this when it was new, you know exactly what I am talking about. If you weren't, you missed out on an amazing experience. It was a very different experience than online PC gaming, or Xbox Live. I can't say what exactly it was, but there was something surreal about it. I put hundreds of hours into this.
9.) 4-player dreamcast games. Rush 2049, Worms Armageddon/Worms World Party and Virtua Tennis were some of the most played.
10.) Playing Dreamcast with my younger sister. We never had too much in common, but she did enjoy playing games with me from time to time. This is my favorite memory of them all. I remember playing Mars Matrix, Rush 2049 and 102 Dalmations with her. There were more, but I can't think of them right now.
11.) Getting Sonic Adventure 2 the day it came out and finishing it within 24 hours.
12.) Buying a second Dreamcast system, just so I could get the Sega Smash Pack.
13.) Hunting for games while the DC was dying. I was always on the look out for good deals on good games, and dirt cheap crappy games. Highlights include $15 for Canon Spike, $10 for Bangai-O, $15 for Gigawing 2, $5 for the Resident Evil games, $2 VGA Box, 2x $5 Top Max arcade sticks with Fighting Force 2 (both arcade sticks were faulty). Seeing the DC go was sad, but it did let me get many more games. Playing games on the VGA box was somewhat of a hassle, but it looked amazing.
14.) Getting Alien Front Online and Grandia II from my sister for my birthday. This happened after the DC was pretty much dead, and prices had fell quite a bit but not much was left. I gave her a list of what I already had, and she managed find two good games.