Hi, and welcome!
First off, $4K is a hefty budget, but any budget should be spent wisely. I'll echo a previous comment and say that I hope you are planning to devote time to learning about exposure, light, and how to use them creatively. Else, you may just be better off with a P&S. Ok, enough soapbox - I'll suggest a lineup based on a $4K budget, followed by the rationale. I'm trying to be pretty comprehensive and so you'll see things that I think will really improve your captures, but that you might not otherwise be thinking about.
Canon T3i - $700
Canon EF-S 17-55mm f/2.8 IS w/ hood - $1150
Canon EF 85mm f/1.8 w/ hood - $425
Canon Speedlite 430EX II w/ StoFen Omnibounce - $300
Eneloop 4 AA + charger kit - $15
Manfrotto 190XProB+496RC2 kit - $200
Canon PowerShot S95 - $370
Canon Vixia HF M40 - $650
Sandisk 8 GB SDHC cards x6 (3 2-pk) - $120
DxO Optics Pro Standard - $120
TOTAL = $4050 (not incl. $35 in Manfrotto rebates); all prices are current from B&H (you might find better deals)
Now, why am I recommending all this?
T3i - Good IQ (same sensor as 7D), relatively small/light for a dSLR. Good choice there, more of your budget should be lenses than bodies.
17-55mm - IMO, the best general purpose zoom for a crop body. It was the first lens I bought (with my T1i), and now, a 7D, 5DII, and 9 L-series lenses later, it's still my go-to walkaround lens for the 7D. Fast, great focal range indoors, IS.
85mm f/1.8 - great for portraits of family, great for low light. Why not the 70-200mm f/2.8L IS II? I have that lens and love it...on my 5DII. On APS-C, it's too long indoors, and for stage performances, f/2.8 just isn't fast enough. IS doesn't help much with action, you need a wide aperture to stop action in low light. The 135mm f/2L is also great for that, but much more expensive and too long for convenient portraits on APS-C. For $75 more, you could get the 100mm f/2 instead, almost a twin but I'd recommend the 85mm.
430EX II - indoors, sometimes you need flash, period. Even at ISO 3200 on my 5DII with the 85mm f/1.2L II, sometimes I need flash (f/1.2 gives enough light, but at the cost of a depth of field about as thin as an eyelash - not kidding). The T3i has a pop-up flash, which is nice if you're a big fan of the deer-in-headlights plus redeye look. I'm not. The 430EX II allows bouncing off the ceiling for flattering light, and also provides a more subtle AF assist (red/IR, vs. the series of main strobes from the popup or 270EX, and you can use the AF assist even if you don't fire the flash).
Eneloops - best rechargeables, period. NiMH also means faster flash recycle times (my 430EX II can keep up with the 8 fps burst rate of my 7D, for several shots).
Manfrotto tripod/ballhead - you might not think you need a tripod, but you do. It's a key to getting sharp shots, and can help you sharpen your skills, too, by slowing you down a bit so you think about your shots. Also, *you* are a part of your family. Don't leave yourself out of every shot. If you get a cheap tripod, it will be inconvenient and you won't use it. IMO, Manfrotto is the best compromise between quality and value.
PowerShot S95 - a great camera that really is small. The small interchangeable body with pancake lenses is smaller than a dSLR, but still not go-everywhere small. The S95 fits in a pocket, has a very good f/2 lens (28-105mm equivalent), and a large sensor (for a P&S, that is, same sensor as the more expensive G12), meaning better IQ and low light performance. It shoots RAW for more post-processing flexibility.
Vixia HF M40 - you mentioned shooting several videos per month. Yes, the T3i has video. So does the GF3, and the S95. But they aren't video cameras. Shooting decent video with a dSLR requires a lot of hardware - a steadycam, external mic, etc., you won't get quality movies by just pushing the button. The GF3 is better, but manual controls are very limited, and the ergonomics are quite bad for video. The M40 uses the same sensor as the top-of-the-line consumer ($1500) and low-end pro ($2K) camcorders (sense the theme? You get a lot of IQ bang for less buck with my recommendations of lower end gear with higher end sensors.). It's a large sensor (again, large for a camcorder), meaning good low-light performance. I have the M41 version, a bit more on-board storage (but I usually use the SDHC cards anyway), and an electronic viewfinder which I like, but could live without. Bottom line, if you want to shoot decent video conveniently, get a camcorder.
SDHC cards - I recommend more and smaller rather than few and larger. That way, you have enough memory for a longer trip, but normally you can shoot on one card, then swap and have a backup on the card while you process the images/movies, then alternate and repeat.
DxO - you'll want to shoot RAW, and thus you need a RAW to JPG converter. RAW is like a digital negative. Yes, files are bigger - but storage is cheap. With a RAW file, you can adjust exposure, white balance, etc., with little or no penalty on IQ. More importantly, noise reduction is *much* more effective on a RAW file than on a JPG, and DxO is much better at it than Canon's free DPP. DxO claims two stops better, and that's been my experience. So, ISO 3200 on the T3i and ISO 800 on the S95 will come out very useable.
Long response, I know...but, my goal was to recommend a complete package, not use the whole budget on body/lenses and have you go over for important accessories.
Hope that helps, and good luck with your decisions!