Congratulations on the new camera. Please post a review for it when you get feel you've gotten to know it well.
The Canon SD-Series cameras all have a battery level indicator on the LCD. I think it only shows three levels, though. The current batch of compact cameras have very good battery life, though. If you just make sure to recharge the battery at regular intervals, I don't think you have to worry about it. I'm charging my SD950 IS battery about once a week. And I use it a lot.
Generally the auto setting will take care of things for you. But with the flash off, in low light, you are going to have to increase the ISO sensitivity if you don't want camera shake to ruin your photos. When you're going to take a picture, depress the shutter button halfway and you'll see the shutter speed and aperture displayed at the bottom of the LCD display. The minimum shutter speed you should be shooting with is probably 1/60th of a second. If the shutter speed is 1/15 or slower, you're going to have blurry photos.
Exploring the scene modes is definitely worthwhile as they give you some more creative control. The auto mode won't always do what you want it to. And it's digital - so there's no film to waste. There's no reason not to experiment!