all of what you put are important (size, easy, etc)
No camera does everything perfectly. Prioritize and it will help you pick.
I'm a big, big fan of editing videos. I can't believe anyone can really shoot enjoyable video that doesn't need editing. That's just my opinion.
If you really want to watch without editing, I'd lean towards one of those camcorders that shoots straight on little DVDs. That will save you some work.
standard probably (sometimes high def)
High def starts around $1,000, so it's out of your price range. It also makes things much harder. Your focus has to be more accurate. You get more noise at low light levels. You can't shoot it directly on a DVD.
I recommend that you look at
www.camcorderinfo.com for reviews and discussions about camcorders.
For a camera, I would recommend a digital rather than film. It's so much easier and cheaper to shoot digital. The only problem is getting a relatively inexpensive digital camera with low "shutter lag." That's the time between when you press the shutter button and when it actually takes the picture. Trying to take pictures of kids with a camera with a high shutter lag is extremely frustrating.
A DSLR solves that problem, but it costs a lot more money, requires additional lenses, and is bulkier. I think it's the way to go if you want the best pictures possible, but most people are willing to make all of the sacrifices that a DSLR requires. If you want to go the P&S route, there are a lot of people here that can help you with suggestions. If you want to go the DSLR route, there are a lot of people here that can quickly fire up a rousing argument over who's brand is better.