First off, I would like to point out that every novel is a standalone story and so you can pick up the series anywhere, without fear of not knowing what's going on.
However, if you are keen to enjoy every snippet and nuance of the developing story arc, or you just like to start a new series at the beginning, then here is the order in which the stories should be read, which is chronological order in which the stories take place... kinda...
1) Unnatural History (novel) - May-June 1997
2) Leviathan Rising (novel) - July-August 1997
3) Fruiting Bodies (novella) - September 1997
4) Vanishing Point (novella) - October 1997
5) Human Nature (novel) - November 1997
6) Christmas Past (novella) - December 1997
7) Evolution Expects (novel) - February 1998
8) Blood Royal (novel) - March-April 1998
9) White Rabbit (novella) - May 1998
10) Proteus Unbound (novella) - May 1998
11) Dark Side (novel) - June 1998
12) Anno Frankenstein (novel) - September-October 1943
13) Red-Handed (The first part of Time's Arrow) - May 1998
(The novella Conqueror Worm is set in 1797 and stands apart as it is not about Ulysses Quicksilver at all.)