The stock lens is fine for pictures, focus is not fast but you can get nice shoots. While you can shot video with it, it is not the best for video. In short, faster and fixed aperture lens, f2.8 or faster are best, under f4 are good. But these are expensive to buy.
You can mount old Nikon manual focus lenses that you can get for about $50 each on eBay with an adapter (they range from $15-80, I'm using a $15 one but I got one for each lens so it does not get removed at all.
The fist thing you need to buy is a viewfinder for the LCD, they range from about $150 to 400. Without that taking video is next to impossible for me, at least for handheld and outdoors.
I've been using the T2i for a while now and I LOVE it...
I bought mine with the 18-55 kit lens and it's a mediocre lens at best. You can get somewhat narrow depth of field with it (especially compared to the HVX), but not nearly as much as you would with a faster lens. It's not great for low light and it doesn't have a great look. It also doesn't have a constant aperture so the exposure will change if you zoom while shooting. It's certainly usable and affordable, but not ideal.
I also have a 50mm prime which I greatly prefer to the kit lens in every way. The Canon 50mm 1.8 is great and cheap, the 50mm 1.4 is a bit faster and more pricey, the 50mm 1.2L is a bit faster still but rather expensive for what it is. You definitely must buy one of them. For me, a 50mm prime and a good wide angle like the Sigma 10-20mm would do everything I need.
Of course, if fixed focal length lenses are too limiting for you, something like the Canon EF-S 17-55mm f/2.8 would be great, albeit pricey.
Other things to note:
get good SD cards (class 6 or greater)
shoot on a tripod or use a shouldermount handheld rig for stability
use a separate audio recorder (Zoom H4N, or I use the Alesis ProTrack) or investigate the Beachtek or Juicedlink audio preamps
It's an incredible camera even with the compromises. You may want to use it for more than just family videos once you get used to it.
2.66 GHz 8-core, ATI Radeon HD 4870,
FCS 3, AJA Kona Lhi
"The creative person wants to be a know-it-all. He wants to know about all kinds of things: ancient history, nineteenth-century mathematics, current manufacturing techniques, flower arranging, and hog futures. Because he never knows when these ideas might come together to form a new idea. It may happen six minutes later or six months, or six years down the road. But he has faith that it will happen." -- Carl Ally
a fast lens refers to the widest f-stop (smallest number) f2.8 is 2-stops faster than f5.6, so you can get the same expose at 1/50 sec, f2.8 and ISO of 200 as a 5.6 lens but the ISO would need to be 800.
Also in the end the wide the aperture that is used the shallower the DoF.
Side effect is fast lenses are normally considered pro lens and normally have better glass and as a whole are made better. Of course all this is why the lens cost $$$$.
They are referring to the maximum wide aperture of the lens, how much light it can gather, so a fast lens to me is anything from f1.2 to f2.8
This type of wide aperture is also necessary to get the shallow depth of field you initially referred to, because as you stop the lens down to a smaller aperture (f16), you increase the depth of field, making more distance come into focus.
The kit lens is also a multi aperture lens, meaning at the widest angle the f stop is f3.5 and as you zoom in, the minimum aperture becomes f5.6.