There are, basically, two ways you can go: camcorder (and that includes things like the GoPro) or DSLR. The difference between the two is that a DSLR lets you use DOF (depth of field) -- you can throw the background out-of-focus, while keeping the foreground sharp. This is one of the things that makes movies look like movies. However, DSLRs tend to be on the more expensive side, and a bit outside your budget.
The GoPro is an amazing device, but it has a fixed focal length wide angle lens, which limits what you can shoot and, as noted above, you can't do shots that exploit DOF.
Also, 4K (and 2K) is still, almost exclusively,
a professional format for shooting films that will be screened in commercial movie theaters. It chews up enormous amounts of data very, very quickly, and is difficult to edit unless you have a very, very fast computer with a lot of memory. HD (1080p) is a far better format for home shooters, though there are many flavors of HD.
1080p specifies the resolution (1920 x 1080 pixels). It's usually expressed as 1080p/60 or 1080p/30, with the latter number specifying the frames per second. Broadcast HD is usually 1080i or 720p ("i"= interlaced scanning, "p" = progressive scanning). Again, 1080p/60 is preferable, but most camcorders are 1080i/60 or 1080p/30. Less expensive DSLRs are 1080p/30, higher-end are 1080p/60. For your purposes, the difference probably wouldn't be noticeable. Another important factor is compression. There are several compression schemes around for HD, but the primary characteristic to look for bit rate. 15 mbits and below won't look good. 18-20 mbits is okay, 20 and over is best (and is available in amateur camcorders and DSLRs).
Camcorders will tend to have better low-light
performance than DSLRs. Low-light
performance is important to bear in mind, because, I'd assume, you're going to be doing most of your shooting indoors and you'll be using available light, rather than lighting the space in which you perform.
Even given that, I'd still consider a DSLR over a camcorder. Shooting video with DSLRs usually means using a tripod, and they're not really intended for chasing-your-kids home video. The lower-end DLSRs are not image-stabilized (and IS lenses intended for still photography don't work well when shooting video). However, you can get very high-quality video out them -- many television shows and commercials are shot with DSLRs, though these are "tricked out" with matte boxes, various kinds of mounts, etc.
Canon recently introduced the 70D, and you can still find 60D bodies for relatively low cost. I picked one up in January for just under $600. Add a $200 Tamron 18-250mm lens, and you've got a basic video package that can turn out very fine video. If you're so inclined, there are third-party firmware additions that let you shoot raw video -- this is important for a professional look, because it allows you to "grade"your shots -- match color and contrast so they look consistent.