In case of 7 full days as a time frame for Nice, Lyon and Annecy I would allocate them 3-2-1 and the 7th is pretty much consumed on transportation and arrangements. The TGV connection Nice-Lyon is swift 4h30 min and Lyon-Annecy by train 2h is easy also.
From Nice, you can plan a day for trips in nearby places either west, east or inland. It is easy to move about by train and/or buses.
www.nicetourisme.com and the numerous Nice forum threads here at TA are good sources for what to do in Nice.
Cheap(ish) hotels with reasonable quality can be found reviewed here at TA, hotels.com and booking.com etc. I am sure you will find a suitable one.
In case you want to travel by bus, have a look at www.lignesdazur.com bus route map when selecting the hotel. Nice-Ville main railway station is on Avenue Thiers, you can locate it easily in google maps. Lots of hotels can be found between the Promenade and the station. Several buses pass on one way streets Rue de la Buffa and Rue du Marechal Joffre. Tramway goes along Avenue Malaussena / Avenue Jean Medecin, turning NE at the very central location Place Massena, which is near the old town, the Promenade and the small pedestrian zone.