I'm really happy with the digital camera (Canon Powershot, 2006 model) I have. It's for photographs mainly, but with a 2GB card it can take up to an hour of high quality video with audio. It's easily on par with the quality you'd get from a ten-year-old video cassette camera, and within my budget at less than $200 (plus a bit for the memory cards).
When I run out of space on a card, I usually make an archiving decision and upload the videos I don't need to carry around with me onto my computer for safe keeping. You could do the same with a digital HD video camera or a photo camera like mine. What you want to think about is what format storing and sharing your videos will work best for you.
It can be expensive using memory cards to permanently store your videos. If storing your videos in a format that's easy to share and view with neighbours and family, you might consider getting a DVD burner. They're cheap enough now that they won't break the bank, and it really is convenient and fun to pop a video into your standard DVD player, relax on the couch, and watch your vacation clips.
But perhaps you share videos over the Internet more than you would on something like a DVD, anyway. Most video and photo cameras sold now come with RCA video and audio output, and considering that most newer TVs also have easy accessible auxilliary video and audio inputs, it can be really easy to connect your camera directly to your TV to review your videos or photos.
I like to keep my favourite videos on the memory card in my camera, and even doing that I still have plenty of room to take impromptu videos when I go out somewhere. If a video makes the cut, it stays on the camera for easy sharing. If it's something I really want to keep or otherwise warrants removing and archiving, I download it to the computer.
How you and your wife want to share the videos you take should be a big factor in your decision on what kind of product to go with. If your wife doesn't feel comfortable using the computer, you might want to work out how you would burn DVDs for her after she's taken her videos. Or maybe she might be comfortable with learning how to connect a digital camera to the TV.