I spent several years attempting available light only indoor shooting at church events with varying success. As is well understood, there just isn't enough light to shoot with all 'keeper' exposures.
The first battle is to hold the camera steady for the 1/30th and slower shutter speeds needed for reasonably 'good enough' exposure. As a tripod is too big to lug around in church, I settled on a monopod with a ball head, and my leaning against a pillar or wall or seated while taking my shots. That helps tremendously.
I soon discovered that I needed faster lenses...f2.8 and faster. Image Stabilization is very desireable, too, to reduce/eliminate my less-than-steady hands from causing blurred pictures at slow exposures, even on a monopod. At f2.8 and faster, I could keep the ISO down to 1600 on my 60D. But there were times I had to use 3200 to get anything reasonable. 'Chimping' almost every photo is a must, to verify you at least have an image that is reasonably exposed and adjusting as needed.
Of course, shooting wide open on fast lenses results in an incredibly thin depth of field. So perfect or very nearly perfect focusing is an absolute requirement. Using AI Servo is critical for keeping sharp focus. Trying to get a group of people all within the DOF is close to impossible. Still at the slower shutter speeds, subject motion is still an issue. So I tried to take my pictures when I expected them to be generally stationary. My in-focus, not motion blurred keeper rate shots was about 15-20%. Not great, but acceptable. It just meant I had to do a lot of 'spray and pray' shooting to get enough keepers.
The 'ultimate' solution was upgrading to a 5D mark iii. ISO 6400 with little noise, and still acceptable noise at 8000. That, combined with fast lenses, I can keep my shutter speeds at 1/125th and faster, pretty much stopping subject motion (and mine!) for most of my shots. My keeper rate is now closer to 75% for exposure-related issues. So I'm more than satisfied.