Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?
Time is infinite, both ways. If it started from some point, then you can always assume that there was still lot of time before that "some point".
I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.
That is our (human's) perception of time. But you can always imagine a mysterious clock which was still ticking while and before big-bang.
Does time have a starting point of note?
No. Time is eternal, hence no starting or ending points.
One may ask, if the time is eternal, then that means we have infinite past. So how come we reach to the present moment?
That seems to be philosophical question, but it can be proven logically as well. For now, we may think that there were infinite cycles. All cycles were identical. Due to infinite cycles, time becomes infinite. Due to all of them being identical, at some point we can exist in anyone of them; because length of cycle is finite.
In other words, for example if big bang was start then big crunch is end of the cycle. That includes all the events happening in between. These 1 finite length cycle would keep repeating perfectly.