Cinque Terre To Monaco ?

Rome2Rio.com suggests the train. You'll probably have to change trains in Genoa and Ventimiglia (the border), possibly also at either La Spezia or Levanto. Running time from Vernazza is likely to be between 5 and 5-3/4 hours. If you bought tickets right this minute for Friday, I think you'd pay between 40 and 50 euros.

Take the train to Ventimiglia and then an additional half hour train ride to Monaco. Regional trains take just over 4 hours to Ventimiglia (1 change of train in Genoa) and cost 19.10 euro. Cost of the ticket from Ventimiglia to Monaco should be fairly inexpensive. There are also some Freccia/Intercity trains to Ventimiglia that cost 37 euro, but they're not much faster (10-30 minutes). The most expensive option would be the Freccia/Thello train combination which would take you all the way to Monaco, with ticket cost closer to 50 euro. If you purchase a ticket all the way to Monaco at the train station in Cinque Terre, make sure the ticket seller understands you want to go to Monaco-Monte Carlo and not to Monaco di Bavaria (Munich in Italian). You can check train schedules at trenitalia.com, noting that some Freccia prices listed my be for discounted and unchangable pre-purchase fares (can purchase up to prior day if available).

Thank you very much for the information, car service seems out of reach price wise (I'd probably prefer it .) I'm going to look into the fast train , hopefully it will be my best option .

You don't say where exactly you are staying at the Cinque Terre.
I tested Monterosso, and the fastest option is the Intercity departing Monterosso at 10:55. One change at Genova Piazza Principe where you take the Thello to Monaco. You will be there at 15:38. The cost is under 50€.
Check for yourself on www.trenitalia.com
The station name in Monaco is: Monaco-Monte Carlo
The departing station is whichever town you are staying at the Cinque Terre.

If you want you can also check on this website, which gives you also the slower/more changes options. Same station names:
https://www.bahn.com/i/view/index.shtml

Thank you for all the suggestions. The train is the most practical way to gon!
Thanks