I'm a novice so don't know whether I need the 18-55mm lens that comes standard with the Canon EOS Rebel T6i or should I instead get the 55-250 mm lens - or both? I'll be using the camera primarily for family photos during holidays and vacations - both indoors and outdoors. On occasion will take longer shots at sporting events and on trips out west.
You will definitely need the 18-55mm lens for shooting indoors and general all around photography as it covers wide-angle to short telephoto. The 55-250mm is a great lens for wildlife or sporting events, but it is much too long for shooting indoors.
There is a 18-135 option like what Rob had mentioned above. When I got my first DSLR, I just had an 18-55 and it would really irritate me at times as it has a very short range of zoom. Also, if you have two lenses, 18-55 & 55-250, you will have to switch between the two quite often and that could a) irritate you as a beginner, and b) more importantly expose your camera and lens to dust!
If you have a budget constraint, have a look at Tamron 18-135. Quite worth the money!