I think you're operating under a few false presumptions, the first being that that all infinities are instantaneous, which they are not!
Imagine I told you to start counting at 1:00 PM tomorrow, starting at the number one, and then to keep counting, once a second, until I say stop.
So at 1:00 PM you'd start: 1...2...3...4...
At 1:05 PM you'd be at: 300...301...302...
And at 1:10 PM you'd be at: 600...601..602...
Ok now I tell you to keep counting to infinity, so that you never stop counting! Eventually you'll reach every number you can think of (and even more, since you can always count another second after the one you just got to), but it might take you weeks, or months, or centuries to get to there!
So in that example we had a start, 1, and we're going all the way to infinity (i.e. we'll never stop). In the same way, the universe could start at one time and keep expanding forever (the only difference is that the universe "starts counting" at 0 instead of 1).
But why 0 and not 1? Or 2? Or -356? Or even -infinity? Why did the universe start 13.8 billion years ago? What about the time before the big bang?
This is a very, very tricky question (and one which may never be answered), but most cosmologists will tell you that there was no time before the Big Bang (since there was no space!), and even if there was, we probably wouldn't be able to probe it.
I will suggest you check out Sean Carroll's blog with a few great explanations to these kind of questions:
http://preposterousuniverse.com/writings/cosmologyprimer/faq.html
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