Chris and I learned a great tip from a Dutch couple we met on the beach: there is an awesome walking path between Nice and Monaco that only takes about a half hour and is so worth it. There are tons of awesome spots to stop for a swim along the way, along with Elton John's favorite beach! Get off the local train from Nice at the Cap D'Ail station, walk down to the water and turn left to head to Monaco (and right to check out Sir Elton's fav beach). These are the stunning pristine waters of the Riviera you dream of.
Monaco has great public transit that takes you all over Monte Carlo. Take the #1 bus to the beautiful hilltop village with stunning views of the harbor.