Paris to Bern

[www.seat61.com] makes a good starting point to read up on each country's rail system and provides links to the national railway operator's website.

I can't find any flight cheaper than $300, and that's outrageous, considering I've always heard how cheap Europe's air travel is
300 Hong Kong $ is a very reasonable price on that route.

And the French site tells you there is a train every evening at 17:57, arriving in Bern at 22:30, i.e. 4 and a half hours. It leaves from the Gare de Lyon in Paris.

Looking at the price, if you book soon for March 13 (I took a March date at random), there is a non-refundable ticket for 25€. Regular fare around 90€.

Paris - Bern:
direct train Paris dp 17.57 - Bern ar 22.30. Fares start at 48 EUR if you book well ahead in the net, up to 146 EUR at the railway station counter.
Other 39 daily connections with one train change at either Basel, Lausanne, Geneva or Frasne.
http://www.tgv-lyria.com/main/FCK/File/site_en/home/home_france_suisse.asp
Fares for https://www.easyjet.com flights Paris - Geneva (4 daily) start at 48 EUR with 1 piece of checked in luggage. Train fare GVA airport - Bern (2 hrs trip, trains every 30 min): 44 EUR
http://www.sbb.ch/en/home.html
1 EUR is actually 1,33333333 USD

To book the train ticket on your Eurail pass, you'll have to pay a supplement. If you have not already bought a pass, I'd consider paying for point-to-point tickets instead, as that's possibly cheaper.

"considering I've always heard how cheap Europe's air travel is"

I suspect you're referring to low-budget airlines like EasyJet, Ryanair, GermanWings, WizzAir, etc. These airlines can offer good reals, but they do not fly every route and do not translate into "cheap air travel" as a general rule for Europe.

Whilst the links the others have provided are solid, it sounds like you wanna hit http://www.bahn.co.uk to figure out travel durations for your europe trip in general (the ones listed above by #3 don't have a europe wide timetable engine).