As a practical means of getting around intramural Paris, it's hard to beat the Vélib' system. If you time your trips carefully, you can travel for a euro a day (less if you buy a weekly pass), locks and lights are included, and you don't need to worry about storage, maintenance or theft (within reason).
There are a few shops around Paris that specialize in used bikes. Bicloune on Boulevard Beaumarchais is a good one (used bikes are in the basement). It used to be that classic city bikes (known universally as "vélos hollondais" whether actually Dutch or not) commanded a premium, but with the Vélib' taking a large part of that market, prices may have become keener. If your French is decent, you might be able to negotiate a buy-back deal.
Paris markets aren't great for used bikes. There are usually a few stolen supermarket bikes at the Puces de Montreuil, but nothing you'd want to keep.
The MDB-IDF (Mouvement pour la Défense da la Bicyclette - Ile de France) organises bike sales in conjunction with some of the Paris mairies
. Prices tend to be quite high, but there are sometimes bargains to be had. If there are any in the offing, you'll find details on the MDB site:
The petites annonces
forum at pignonfixe.com is quite busy:
As well as the sites bduc61
mentions, trocvélo and vélovert can be good places to pick up used gear:
If you're around in the autumn, there are two big bourses aux vélos
within a short train ride of Paris (at Baillet-en-France and Puiseaux) that offer excellent opportunities to browse and haggle.
Look out for posters for local vide-greniers
. There are often a few bikes and sometimes wheels and other parts among the junk.
Don't forget that older French bikes (late eighties or before) have obsolete French (metric) threads, and replacing parts in the longer term could be a tricky proposition if you take the bike home with you.
Calmes dans le demi-jour
Que les branches hautes font,
Pénétrons bien notre amour
De ce silence profond.