Dinoysos are my favourite - very eccentric and arty, with influences ranging from the Pixies to Tim Burton or Salvador Dali. They have some songs/lyrics in English, but really it is their French material which showcases their style the best. I would say their most approachable album is "Western sous la neige" (which was produced by legend Steve Albini). Check out this:
https://www.youtube.com/watch?v=XdSbSBROdnk for a little taste of it.
Indochine is an older band (they started in the 80s with some new wave hits) but their recent stuff is definitely very modern and with the album "Paradize" (2002) they became a staple of French music all over again. Their most energetic songs have a very punk/electro vibe - although I like that, I prefer the mellow side of them, which I find just plain sweet and charming:
https://www.youtube.com/watch?v=6_vYYb7S6qI
Other bands of interest include Noir Desir (mentionned before) who were THE French rock band of the late 90s - and despite the scandals it still sounds great, very clever and politically engaged:
https://www.youtube.com/watch?v=J8Z549GKkeM . You should probably also check out Louise Attaque - they split up in 2001 I think, but their style was original, mixing modern and traditional music. The result is not to everybody's taste (I personally love it) but they're worth looking up for the quality of their lyrics:
https://www.youtube.com/watch?v=0emXuhPTWUM. Same goes for Matmatah who mixed up modern rock and celtic music from their native Britanny:
https://www.youtube.com/