It's interesting that this came up (again) right now. I've just spent the past week finally digitizing some old VHS home movies that turned up in my parent's basement. We just never got around to doing anything with them, and VHS players may become fairly rare in a few more years. I still haven't snagged a cheap Beta player so the movies from my earlier childhood are mostly locked away for now.
Back in the day, tapes weren't exactly cheap. Nor did many people own a video camera. So mostly what we have is a 2 hour tape of a single weekend, taken with a borrowed/rented camera. Then another tape from several years if not a decade later. The quality of the media, plus degradation over time, means some of it looks pretty awful. You can mostly make out who people are but it can be tricky. Etc. I've also started looking into scanning all of our printed photos, of which there are many more - everyone owned a camera in the 70s/80s - but still, maybe 5-10 pictures for any given event or day, and many things were simply never photographed because they didn't seem important at the time, or we ran out of film, or whatever.
And looking through all of this makes me realize how precious these relative few records of my past are. There's maybe 2 hours of video with me in it and a few hundred photographs, spread over decades with large chunks of time completely missing. So when I look at these things, it's remarkable. Some of it I haven't seen in years, some I've simply never seen. I'm at the perfect age where it's not completely unheard of to have video of one's self, but it certainly wasn't common nor made in quantity - so you take what you can get, and there's a sense of fascination with it.
I contrast that to kids growing up today. Damn near every single day of their lives will be recorded, in high quality audio, video, and images. By the thousands of hours and tens of thousands of pictures (hey, digital storage is CHEAP). They will continually be exposed to it, if my friends and family are any indication - some of them constantly re-watch videos of first birthdays, first walking, first vacation etc etc etc. For most of my life I've had to rely on memory alone, with a few pictures to remind me of what any given house looked like, or the yard, or my friends at the time, or what have you. This next generation will have it in their face at all times, and accessible throughout their lives.
Just got me philosophical, I guess. I'm completely fascinated that video of me even exists from when I was 10. My nephews right now have a hard time understanding why we don't have video of their dad through every single month of his life.
As for storage, I'm digitizing everything to whatever open and widely readable format works that has enough quality considering the source material, keeping it on hard disk (backed up to another), and sent out to several family locations on burnt DVDs. Within a few years the space will be almost trivial and I'll probably add a backup to my keychain. But my entire recorded life can be stored in a few gigabytes. Your kid's first week probably contains more. I think what I'm hinting at is that you might want to consider not keeping every single last video and photo if it becomes too much of a burden. If there's less around, it will become all the more precious and fun to look at in the future.