How to expand borders?

Libraries generate more culture (2 cpt vs. 1 cpt for Temples). You can, however, have up to seven temples per city (one for each religion). But build Libraries first if you want fast expansion.
 
Depends what you mean by "fastest".

If you mean it's very early game, your new city only has its first ring available, 0 culture is being produced, and there's a critical resource in the second ring (the full "fat cross"), then fastest way to expand that border--as in available before any other method--is to discover Mysticism and build an Obelisk for its +1 culture. Libraries require the Writing tech, which has a couple of prerequisites, where as Mysticism is a starting tech--no prereqs and your civ may even start with it.

Alternatively, with Mysticism, you could build Stonehenge, which has the same effect for all cities without bothering to build any obelisks. This tends to make Stonehenge one of the more desireable early wonders. (Long term, though, these options aren't viable; Stonehenge and Obelisks are made obselete by a relatively early tech, Calendar.) You could also play as a leader with the Creative trait so all your cities produce +2 culture right off the bat.

On the other hand, if by "fastest" you mean "what will build me towards each border expansion sooner over the course of the game", then it's whatever buildings will produce or magnify culture, which is available in their description when you hover over them. Temples add +1 culture, Libraries +2, so the latter will build up towards the next border expansion faster.
 
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