I need to write a paper on French phonology for my Phonology class so I was wondering if you could give me some advice on where to start? I'm mainly looking for textbooks either in English or in French but online resources are also fine. In each book I've opened, it seems like every author has their own classification of vowels and consonants and I don't know which classification would be the most accurate from the phonological point of view.
I think the answer largely depends on what you mean by "phonology". Since you specifically mentioned "classification of consonants and vowels", I assume your main interest is in taxonomic of the segmental inventory. In that case, accuracy is not really a relevant consideration, since classifications are rarely "inaccurate". Classifications differ because one follows theory A and another follows theory B, and the dispute is really over the classificatory framework. So for example, Jakobson & Lotz 1949 "Notes on the French phonemic pattern" presents one analysis of French phonemes. You can compare that to Schane 1968 French phononology and morphology, Fougeron & Smith in their French Illustration of the IPA, and there is also a general book by Fagyal et al. French: a linguistic introduction where you can an analysis of the inventor; Charette 1991 Conditions on phonological government will contribute a different theoretical perspective. C.D. Walker also analyzes French phonemes, in standard and Canadian French. Vol 25 issue 5 of Linguistics was a special issue of French phonology and phonetics.
Some of the most recent, comprehensive and descriptive works in English about the phonology and phonetics of Modern French are Phonological Variation in French: Illustrations from Three Continents (ed. Gess, Lyche & Meisenburg, John Benjamins, 2012) and French Sound Structure (Walker, University of Calgary Press, 2001).
French: A Linguistic Introduction (Fagyal, Kibbee & Jenkins, Cambridge University Press, 2006) also covers phonology and phonetics at some length. Dialectal variation of French around the world is documented in great detail in Varieties of Spoken French (ed. Detey, Durand, Laks & Lyche, Oxford University Press, 2016).
If you are looking for the phonological inventory, here some French literatures:
MARTINET, André / WALTER, Henriette, Dictionnaire de la prononciation française dans son usage réel, Paris : France Expansion, 1973.
WALTER, Henriette, La dynamique des phonèmes dans le lexique français contemporain, Paris : France-Expansion, 1976.
WALTER, Henriette, La phonologie du français, Paris : P.U.F., 1977.
Gougenheim Georges, 1935, Éléments de phonologie française, Étude descriptive des sons du français au point de vue fonctionnel, Strasbourg, Public. de la Faculté des Lettres.