There are direct trains (TGV Lyria) but they need a reservation with an extortionate price, especially in first class, see this post:

A much cheaper way is to book a TGV to Strasbourg or Mulhouse, then a regional train to Basel and then an intercity to Zurich. There are also possibilities to do the whole journey without reservations but that will take a lot more time.

What the best route is, depends on what you're looking for.

Thank you!  That helps.  It looks like it saves some money and gives us more options/flexibility.

Do you know if you happen to purchase reserved seats and then are unable to use them for the scheduled time, can you get a refund or use them on a later train?

Reservations can only ever be used on the booked departure. In most cases no refund is possible, this does depend on the train company and who you book with.

For that journey only the TGV Paris to Strasbourg or Mulhouse needs reserving, the rest of the trip and indeed practically all trains in Switzerland do not need reserving

Great advice has already been given. I'll only add that there are 2 more options if you want to travel through the Jura mountains :

- TGV Paris-Gare de Lyon - Frasne 07:53 - 10:42 (also 11:53 and 17:47 departures) limited 10€ or 20€ (like all other French TGVs)

- TER/RE Frasne - Neuchâtel 10:53 - 11:52 guaranteed connection in Frasne

- S5 Neuchâtel - Bern 12:01 - 12:52 or you wait half an hour for the much fancier IR66 12:32 - 13:07

Doesn't take more time than via Mulhouse or Strasbourg Until a few years ago direct TGVs to Bern used to take that line !

Or via Belfort-Montbéliard TGV. Here is one of the connections :

- TGV Paris-Gare de Lyon - Belfort-Montbéliard TGV 11:13 - 13:46 limited 10€ or 20€

- TER/RE Meroux TGV (station above the TGV tracks) - Biel/Bienne 14:03 - 15:41

- IR65 Biel/Bienne - Bern 15:52 - 16:18