A common view now in the physics community is that the Universe is infinite in extent (which makes Hubble sphere an infinitely small part of the Universe). On the other hand, there is the Big Bang theory which states that the Universe began about 14 billion years ago. So, how can the Universe be both infinite and have a beginning? Or does the Big Bang only apply to the Hubble sphere?
The solution arises during the inflationary epoch. It is hypothesised that some sort of phase transition during the first moments of the universe drove a tremendous expansion/inflation of space by many, many orders of magnitude. A few details are in the link. I'm not sure that an "infinite" universe is the prevailing view; just very much larger than the observable universe.
The thing is that this faster-than-light, exponential expansion of space effectively isolates different parts of the original universe meaning that our little patch is just part of a much greater whole. It simultaneously explains why the universe is "flat", and highly isotropic and homogeneous, and most importantly for this question, addresses the "horizon problem".
The first section of the wikipedia entry on the exponentially growing scale factor during the inflationary epoch would be useful reading.
If the Universe is infinite, then it had instantly grew from zero to infinity at the moment of Big Bang.
Note, that this is a possible model from General Relativity (GR).
But most probably GR is not applicable at the moment of Big Bang, because it contains singularity.
First of all, the universe is not infinite. There is enough reason to disprove an infinite universe model. In such a case, the universe can have a beginning. Reasons to why the universe simply cannot be infinite are as old as the time of Isaac Newton. The problem is gravity. In an infinite universe, the total gravity of the mass it contains would be infinite and the whole universe would collapse into a space of infinite density, a black hole. On the case of expansion, modern theories such as of inflation, accurately describe certain features what we observe, such as the CMBR (Cosmic Microwave Background Radiation).