I will be using my Eurail pass to travel from Geneva to Bern. Once I reach Bern, can I use the pass for travel within Bern on trains, trams, buses etc?
No, urban transportation is not covered by Eurail passes. You can select your type of Eurail pass in this overview and read the "Overview" and "Where it's valid" sections for details.
Will you be staying in a hotel in Bern? If so, you can make use of the Bern ticket, offering free urban transportation: www.myswissalps.com/be rn/travel. If not, you should just purchase regular tram and bus tickets.