The Neuchatel commune office has a brochure which will list all the places you can go to to pick up French. Anyway, this is the list I got from that brochure - I kept it even though I have already signed up for classes.
Littoral neuchatelois
Lycee Jean-Piaget
Rue des Beaux-Arts 30
2000 Neuchatel
032 717 88 00
Ecole-club Migros
Rue du Musee 3
2001 Neuchatel
032 721 21 00
Ecole Benedict
Rue de l'Ecluse 38
2000 Neuchatel
032 725 29 81
Interlangues
Grand-Rue 1a
2000 Neuchatel
032 724 07 77
Centre de formation professionnelle du Littoral neuchatelois (CPLN)
Passage Max-Meuron 6
2000 Neuchatel
032 717 78 40
Inlingua
Avenue de la Gare 16
2000 Neuchatel
032 724 62 29
Universite de Neuchatel
(Institut de langue et culture francaise)
Avenue du 1er Mars 26
2000 Neuchatel
032 718 18 00
Globe-Trotter
Verger l'Ecuyer 8
2068 Hauterive
032 753 80 50
(This is near Neuchatel - 15 mins by bus)
I've actually signed up with Ecole-club Migros because the fees are affordable and it's next to Place Pury so it's convenient for me to either take the bus or cycle there.
Anyway, hope this info helps.