I'll start with a short answer followed by an elaborate justification. I would recommend a Canon HV20 with an external firewire hard drive recorder. This is based on owning and enjoying a Canon HV10, Sony HC3, Sanyo HD1, Canon TX1, and Aiptek A-HD high definition camcorders, and miscellaneous SD cameras.
There are many measures of quality, but one of the most basic is resolution. Here are some picture size approximations for reference: SD 480I = 720 x 480 pixels, or about 350K pixels per picture (two interlaced fields); 720P = 1280 x 720, or about 1 megapixel per picture; and 1080P = 1920 x 1080, or about 2 megapixels per picture (1080I requires two successive fields for one picture).
And here are the pixel rates per second for various common formats: SD, 10 megapixels per second, 30 complete pictures; 720P30, 30 megapixels per second, 30 complete pictures; 1080I, 60 megapixels per second, 30 complete pictures; 1080P24, 48 megapixels per second, 24 complete pictures.
Interlaced standards imply that a picture will be reproduced using two interwoven fields, the first consisting of the odd numbered lines, the second of the even numbered lines. Most cameras expose each field separately, 1/60 of a second apart, which means that moving objects in the image won't line up when a picture is composed from the two fields. The benefit is that when they are shown as video, they produce a smoother moving image. Progressive standards imply that the entire picture is exposed at one time, as it would be on film. 1080P24 is supposed to emulate the appearance of film, as film is typically exposed at 24 frames per second.
We'll use this information below.
The first consideration in your choice of equipment should be the value of the content over your lifetime. It takes little more effort to shoot good video using HD formats versus SD formats, but results in pictures with three to six times the resolution. As resolution standards increase in the future, they will have greater value than SD video of the same content. It doesn't matter if this is personal or professional video, the same rule applies. The cost of the equipment is far less important than the time and effort you will spend using it over its life, and the content you will produce -- go for the best results.
Within the HD camcorders you are considering, there are four predominant compression formats: MJPEG (Canon TX1, JVC), HDV (HV20, others), MPEG-4 (Sanyo, Aiptek), and HDAVC (really MPEG-4 AVC, aka h.264). Real AVC requires lots of horsepower, and current implementations leave room for improvement, as one might expect in the first generation. HDAVC cameras typically record to hard disk or flash media.
MPEG-4 is generally for flash media cameras, such as the HD1 and A-HD. It allows 720P30 high definition recordings around 4 gigabytes per hour with acceptable quality, although quite a bit short of HDV quality. This is necessitated by the current capacity and data rate of flash memory cards. Memory cards still cost about ten times as much as DV tape for the same storage capacity, so you will probably want to erase and reuse the cards. Just be sure you have made two copies to different storage devices before erasing the original.
MJPEG is a format that offers easy editing, much like DV format (each picture is compressed individually). However, it eats memory like crazy (Canon TX1 uses about 8 GB/hr). And the difficulty of editing MPEG formats which share information from picture to picture is only a problem for the guy who has to write the editing program -- for you, there should be no additional burden.
MPEG-2, which is the basis for HDV, is well understood, and although it isn't as efficient at compression, produces an excellent image. HDV compression puts an hour of high definition content on a one hour DV tape, the same as a standard definition DV camera. HDV editing tools are common. DV tapes are inexpensive enough that it makes sense to never erase them, keeping them as your archive instead.
Speaking of media, DV tapes store about 12 GB for about $4. Flash memory is down to about $5 per gigabyte, and hard drives are around twenty cents per gigabyte to forty cents per gig for tiny drives. While hard drives are remarkably reliable, when they fail, their entire contents is gone. With video tapes, only the section affected by the failure (e.g. crinkling) is lost. When video is captured to a hard drive, it is ready to plug into a non-linear editing system immediately, avoiding the "real time" (1x) transfer of content from tape to disk before editing can begin. An ideal capture medium in terms of reliability, cost, and workflow is a removable hard drive.
But wait -- none of the consumer camcorders have removable hard drives! So you have to take the camera out of service in order to edit the content of its hard drive, or at least transfer it to another drive before editing, since you'll have to erase the contents of the camera's fixed hard drive before continuing to use it!
What makes a lot more sense is to use an external hard drive, capturing simultaneously to DV tape for archival purposes (copy 1) and to the disk (archival copy 2), which can then immediately be used for editing. Unfortunately, external firewire disks (with controller intelligence to be able to download) are remarkably expensive -- about the cost of a portable computer. So -- use a portable computer! Run an editor with a built-in capture program to record the content as you shoot. If you record it to an external hard drive, you can hot swap drives in the field for immediate editing.
The Canon HV20 is about the best quality, most versatile HDV camera available (according to reviews from owners at Amazon, NewEgg, CNet, etc.). My HV10 has been very pleasing, and the HV20 offers both 1080I and 1080P24 recording formats. It is said to have an actual 1920 x 1080 sensor (many HDV cameras are 1440 x 1080), with reasonable sensitivity and noise. It has an external mike in, and a headphone output (it is very important that you monitor your audio while shooting).
So that is the basis for my recommendation. However, I also recommend that you get an Aiptek A-HD (about $120 on sale) to play with, so you'll know what is coming. I have always associated Aiptek with gimick cameras that were fun but low quality. The A-HD is a real surprise in quality and low noise, and you can afford to give an HD Camcorder to your children! It seems to readout the video sensor in real time, so that if you pan quickly vertical objects tilt. Getting good audio is difficult. You'll need a tripod. But by and large, it is really cool. It can even record an NTSC video input (with audio) and work as a portable video (or MP3) player.
I've carried the HD1 (pocket sized) or the TX1 (even smaller) at all times for a few years. The world's finest camera setup won't do you any good if you don't have it with you when an opportunity presents itself. The 720P30 format is just fine in this context. The A-HD could also serve in this capacity.
By the way, great video requires great audio. Small portable recorders with good microphones (such as the iRiver IFP 7xx and 8xx series sold as MP3 players, or the Zoom H2) have crystal controlled timebases, just like your digital camera. As a result, the audio can be synchronized during editing and (as long as all devices remained rolling during record) they should remain in very good sync. You can sprinkle recorders in areas where you expect interesting audio, and once everything is rolling, clap once in front of the camera so you can line things up later. In this way, you don't have to be dependent on the camera's mike, or worry about dragging around the external microphone wire while you shoot. And you are not limited to the single channel of audio a wireless mike typically provides. If you get a wireless mike, remember you want a lightweight, battery operated receiver (most come with clunky AC powered receivers).
Although your camcorder will be great for handheld shots, please also get a decent tripod, with a quickly removable camera mount (and a spare platform), with "fluid head". The fluid is a viscous dampling fluid, which almost guarantees all of your camera moves will be smooth. A jerky tripod is almost useless for a video camera. Also, when you shoot on the tripod, turn off the image stabilization which will otherwise produce strange lagging movement effects.
Please remember when you are shooting that camera moves and zooms are like fonts on a page -- use too many and you distract from the content by calling attention to the camerawork.
So the answer, although it can be boiled down to a rational choice of equipment, is not intuitive. First, the cost of the equipment will be of far less importance than the value of the content; second, great video requires great audio; and third, great video requires great attention to lighting.
One last thing: Please back up your media, both locally and on distant servers. There are services which specialize in this, but with a little effort, you can use private space on a shared Internet host that is readily accessible but secure. Hosting space in a distant city can cost as little as $6 per terabyte(!) per month, but guarantees that any local catastrophes won't destroy your family picture and video collection. And if you wish, you can also set up your own website in the public space.
Whatever you choose, Russ, I hope you have a blast.