Dear Namesake,
So I have heard the train cost is 0.08 CHF per minute, is that true even for me as a foreign US citizen?
I don't know where that came from. Train tickets are priced by distance. And your citizenship makes no difference to the price (why should it?).
There are some discounts available to temporary visitors (short-term) and others designed for residents (long term). For tourists the two main offers are:
Tourist Half Fare Card (CHF 120 for 1 month): See here: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-halffare-card.html
Swiss Pass: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-travel-pass.html
You DO NOT need to buy tickets in advance, nor make reservations. Normal tickets, which you buy at the station (from the ticket machine or manned ticket counter if there is one) are valid on any train on that date.
How easy is it to get from place to place? How are the train times? Do trains run often?
Very Easy. There are 1839 rail stations in Switzerland. Every single one of them has at least one train in each direction per hour, often 2 per hour. Plus feeder buses to and from the stations, connecting with the trains.
It is designed to be a "turn-up-and-go" service. You just walk into any of the 1839 stations at any time, buy a ticket to any other of them, and you are on your way in less than an hour.
For times and prices, use the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en
Here is a rail map of Switzerland. Red lines are rail (nearly all rail lines shown, obviously only a selection of stations). Yellow are feeder buses providing an in-fill. Lots more local buses not shown.
Map: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf
What about trains from Geneva to southern France?
The High Speed trains, mostly to Paris, are run by a company called "TGV-Lyria". Website: https://www.tgv-lyria.com/fr/en
Timetable for 2020 is here: https://www.tgv-lyria.com/sites/default/files/inline-files/LYRIA-Timetables-15.12.19_12.12.20-EN.pdf
There is one daily direct train Geneva to Marseille, in July or August only. You can get anywhere else in the south of France by changing trains, just choose your destination and look it up on the above-mentioned SBB website.
I hear there is some sort of pass you need in order to get access to the parks, specifically Grindelwald/ Cable car/ hiking/ overlook. What is the most worth seeing and what are the prices like?
Don't know what you mean by "parks". Most of the mountains are public land, anybody can walk anywhere, no charge. There may be animals grazing (cows, unfenced).
You have to pay for transport (cable cars etc.) Most tourist areas have a local transport pass to save you buying individual tickets. Info on that will be on local websites.
For Grindelwald see link below for Jungfrau region.
Anything else I should know about traveling to Switzerland? What's your favorite part about this country?
I think I've covered most of it.
Favourite part, Jungfrau region of the Berner Oberland. Local tourist website: https://jungfrauregion.swiss/en/summer/
Important question: What time of the year are you visiting?