Aaron, in reply to your question, "What does a noob need to do?", the answer would be to shoot in RAW capture mode, at ISO 800 or so with your camera, and set the lens aperture to around f/2.5 to f/2.8. In a shot like the one of three friends all equidistant from the camera, you could set the lens aperture to a wide opening, like f/2.5 let's say, and still get a hand-holdable shutter speed.
As you've noticed, the default camera-selected exposure would favor a bit more depth of field, with the aperture in the f/4 to f/5-ish ranges, which gives more depth of field and a bit more margin for focusing errors.
Another option would be to use just a very small,small amount of flash, which would boost the lower values; look at the guy's black shirt on the left,and you can see some color noise called chroma noise; if you had even a 1/16th power flash pop to go with that exposure, the entire image would have been "lifted" up above the noise floor.
By tiny amount of flash, I mean setting the flash to Minus 2.7 stops, or to manual flash and as little as 1/16 power. At such low levels, the flash will be a minor portion of the exposure; "most" of the exposure will be made by the lens aperture and the shutter speed, and the flash will provide a lot of help to the darkest values, which is where the chroma noise is worst. Please note---I am suggesting working at ISO 800--the D60 at ISO 1600 is well past its acceptable noise level for most people. It would be far better to shoot at ISO 800, or even to shoot at ISO 400 in RAW mode and underexpose a 400 ISO capture and "rescue" the underexposed ISO 400 shot in post processing by lifting the exposure in software. The other thing is that 1/60 of a second is really not a slow, indoor speed; if you want to shoot indoors in low light environments, the short focal length lenses like 35mm will allow you to work at speeds more like 1/15 to 1/25 second, in that range. it might not seem like much of a difference, but ISO 1600 and 1/60 second are both on the fringes for your current situation. Sure, at slower speeds like 1/15 second, more shots are lost due to camera shake or subject motion blur, but the difference is two full f/stops....ISO 1600 to ISO 400 is two f/stops' difference, so you kind of have to compromise one thing for another at the fringes. If your images at ISO 1600 look too noisy, you need to give reduce ISO level...to get the apertures you want, you might have to lengthen the shutter times. You might have to open the aprture up to f/2, or f/2.2 or f/2.5, where depth of field is shallow and focusing becomes critical. On single-person shots, depth of field at f/2.2 is "manageable"---not great, but not that bad either.
What you kind of have to do is to find the low limit of your own hand-holding ability; if you can practice breath control, and taking advantage of found supports like chair arms, tables, bracing against walls or door jambs, etc, and practice tucking your arms in and supporting the camera with your left hand pressed firmly under the lens, and with the camera strap wrapped tightly around your right hand, it's possible that after some work, you'll be able to hand-hold the 35mm lens well enough to get 75 to 80 percent keepers with the shutter dragging as slowly as 1/8 second. And speaking of that--if you set the camera to Continuous shooting mode, the second shot, or the third shot triggered from one press and hold of the shutter release button can be the most shake-free shots. When shooting on the "fringe" of your gear or own hand-holding abilities, shooting in three-shot bursts is a very productive way to ensure getting a good shot.
