Betty:
If it's bike paths you want, you have LOTS of options. Check out my website at nyrides.freehosting.net Many of the bike paths listed are accessible by public transportation; even the ones out on Long Island. In some cases, I have listed public transportation options. In others, you might have to e-mail me for info.
Regarding the Bronx River Bikeway, it's one of my favorites; however, I would take Metro North to Valhalla and ride south to Eastchester to catch a train home. Stop in Scarsdale for an outdoor deli lunch.
Good luck!