Well, what do we see? When we look at distant galaxies we see something called spectral redshifts -- doppler shifts in the spectral emissions of well known elements. Everywhere we look these objects seem to be moving away from us. How odd. The closer ones are moving away slower, and the farther ones are moving away faster.
If you project those motions backward, it is inescapable that we came from a central point -- a "big bang."
Well, how long ago? We know the rate of speed because the doppler shift obeys specific physical laws. We didn't initially know the distances.
One of the earliest methods was to use the brightness of a certain class of variable stars, called Cephids. They could use paralax, triangulation, on close Cephids and get a correlation between their rate of variation, their brightness, and hence their distance. This seemed a good plan because the Cephids were remarkably uniform in that regard. They were a good measuring stick.
So that was one of the early estimates of the size of the universe. But eventually a second class of Cephids were discovered and all the brightness/rate variaition/ distance calculations had to be recomputed.
Those are the two early methods I can recall off the top of my head.
So we haven't seen anything further out than these most distant objects, and they all have that receding redshift indicative of being sourced from the central big bang. So that puts an upper limit on the age of the universe and the Cephid (and now other means) puts a limit on the distance.
There is no evidence that the universe is infinite nor more than 14 billion years old. I won't speculate beyond the limits of evidence.
The big bang theory and the expanding universe may be strong possibilities given the research and what we know. But does current technology prevent us from seeing out farther than we do now? Could it even be possible to discover galaxies 100 billion or more light years away? This is why I'm skeptical of any claims to age, because there's so much out there we don't know yet.