A camcorder is going to be the easy option and cheapest option. Point, shoot, edit, upload. Simple as that. A DSLR will be expensive because you will fall in love with the look. Then you will discover that different lenses will offer you more creativity, so you will want more. Then you realise how expensive the lenses are (often more then the camera itself)! But if you shoot on DSLRs right the quality will be sensational. I shoot on 5Ds, 7Ds and a 60D. The clip below is entirely shot on the 60D
I have a d5000 with a Nikor 35mm prime and a Nikor 18 - 200 vr. I also have a canon vixia hv30. I say if you got the money get both. When I want good video of my kids I use the HV30. I can't tell my kids to do a re shoot on plays, baseball games or birthdays. I'm no pro so when I need reliable video and auto focus for those precious moments I go to my HV30. When I'm taking photos of my kids with my d5000 I have the option be it 5min to record if a great moment comes up. Also when possible I will have my HV30 on a tripod with my wife or family member working it and me on the dslr. Again If you got the money get both.
There isn't actually anything wrong with a higher end consumer camcorder, you just have to know which is the right one to choose. A lot of them aren't as good for several reasons, i.e. compression quality. But if there are any high end consumer camcorders that are exceptional that I can recommend, they are: Canon HV30, HV20, HG10, HF100. All of these will set you ~$500-1000 (and no more), even less if second hand (~$300). Nothing beats these; these are without a doubt the best higher end consumer camcorders for price, and are extremely popular with low budget indie film-makers. I was initially going to choose a Panasonic but the quality was just below and the price wasn't down there either. Sony was out of the question, the compression quality on most Sony consumer camcorders is quite.... crummy.
I took the direction to go camcorder earlier this year and I don't regret it. Granted, I'm more of a video guy and not really a photo guy at all. I admit that I'm a bit behind in terms of the bigness and greatness of an image that a DSLR might provide., but as for how much behind.... not much at all.
A combination of these camcorders with extra equipment (the Canon camcorders I recommended are all very well expandable - i.e. can take a wide angle lens, mic, etc.... I've even seen someone take an HV20 and turn it into an excellent prosumer rig with several modifications, with exceptional final image quality. A camcorder to start with or with smaller amount of accessories can be smaller, lighter and even cheaper than DSLR based setups. Going camcorder is not a bad choice for the price. A small amount of money can go a long way.
I more recently added a Digital Concepts 0.45x lens (which is inexpensive but still excellent wide angle lens) and an Audio Technica shotgun mic; I'm also about to buy some filters for lighting occurrences. Total cost: under $400 for everything. Value-for-image: Exceptional. Probably behind what a DSLR can do, but not far behind at all.
Here is my original test video with a stock Canon HG10 camcorder, from back in September (AVCHD 1080p, shoots 24p; set me back just $300 back in late September). This was before I got the wide angle lens and the microphone.