The car is only useful if you want to go off the beaten track and visit places not easily reachable by public transport. If it’s Bergamo town you want to visit (definitely worth it), then you can easily get there by train or by the bus already mentioned.
No problem renting in Genova and dropping off in Bergamo, the price difference is not significant.
Have a look at this site to get an idea of what your options would be
There are frequent departures all day by train from Genova Piazza Principe train station to Bergamo taking from around 2h 30 with one easy level change of train at Milano Centrale and currently costing from €18,40. Times are currently online at http://www.trenitalia.com/tcom-en only to June 9, but will be the same or similar after.
Driving will take from 2h 45 plus the time taken to go through the formalities to rent and drop-off your car, and will cost around €37 for the cost of fuel and tolls, plus the cost of renting the car, plus the cost of acquiring an International Driving Permit if you are a non-EU citizen and do not already have one.Edited: 16 May 2018, 09:00