I went from Nice to Monaco a few times. Once by taxi (we went in the evening and the trains end around midnight I believe) - it cost about US$40-50 each way for a full cab. It took about 40 minutes to get there and 25 to get back (we were returning at 1 am, no traffic, and we didn't take the seaside route, and I believe we were going very, very fast.
The other time I went on a Saturday afternoon via train from the Nice main train station. It was very painless, the train was mostly full but not crowded. It was probably about US$10 round trip.

On our first trip, we went into the Monte Carlo Casino (late at night). When you go in, they ask to see photo ID while they collect the entrance fee. I didn't have my passport on me, and thought I was screwed for sure, but the woman said that a driver's license was sufficient, and I got in with that!

Monaco is very, very nice to walk around. You can come out of the train station right next to the marina, and walk around the marina (make sure to see the amazing huge yachts), up to the shopping area.

Another day, I took the train from Nice to Cannes. That is also a very nice ride (perhaps more scenic than the ride to Monaco). Cannes is also a spectacular city, and well worth a visit.