While I must admit that this isn't exactly my field of expertise, as a Casanova aficionado myself I'd suggest you to start by reading his very own memoirs first - the aptly-named Histoire de ma vie that is. There are currently two main editions, the most modern and reliable of which are a three-volume critical edition published by Laffont and another one by Gallimard; both are based on the original manuscript, now stored at the Bibliothèque nationale de France.

I'd recommend the former which includes a wealth of useful notes/critical commentary as well as a handy appendix containing a table of the various currencies of the time, a section devoted to the card games Casanova used to play, a list of the book's countless characters, etc., etc., etc. The Gallimard edition is on the other hand a pricey reproduction of the manuscript as it is, with all its defects and without explanatory notes of any kind what-so-ever.

Should you ever want to peruse the handwritten original, you can find it on Gallica.

 

EDIT: should've warned you that both editions are in French! I don't know of any reputable English translation.

EDIT/2: do avoid the older editons as they're all based either on the relatively faithful - if abridged - 1960 Brockhaus-Plon text (at best) or some 19th-century bowdlerisation. You probably don't want any of these.

EDIT/3: as far as the Laffont edition is concerned do not confuse the one of 2013 with that of 1993, which is a partial revision of the Brockhaus-Plon. It's a woefully incomplete mess with plenty of misprints and assorted shenanigans!