Hello Rayscuba, hope I can help you a bit. If you go indepenently you have I think 3 options.
1) Taxi these are expensive and will charge so about 40 to 50 euros to go to Eze.I believe there is also one taxi based in the village itself but things change so check first if you were hoping to use this to go back to nice.
2) Train this does not actually get you to Eze it goes to Eze Bord de Mer and you then have a choice...you can catch a bus to Eze from there or for people who enjoy exercise there is a 1500ft climb along a rocky path to Eze village the tourist board say can take about an hour and a half but people used to fast walking can do in about 50 mins ( so I am told
it will take less time going down and many people after visiting Eze by bus decide to walk down and either catch the train back to nice or catch the bus back to nice. You will need to check if this is still correct.
3)Bus if you go to nice bus station (gare routier)you can take the bus 82 or 112 to Eze village costs about 1 euro 30 each and takes about 25 mins. I believe one of these buses will also take you into monaco after you have spent some time in Eze.
I suggest you find the bus timetables online & for more information about Eze,I am sure there are some forum members who will help out and also make sure the stuff I've told you is ok.