"My question is, would it be best to buy tickets in advance on a site like Rail Europe, or buy them the day off at the rail station"
Usually the best option is neither of these. For your routes, if you can commit to non-refundable and non-exchangeable tickets, you can save a bundle and buy them in advance on the websites of the train operators (NOT Rail Europe, which does not operate any trains). If you do not know your plans, then you can buy at the station, but you'll pay more, and on some routes risk sold-out trains. However, other trains do not sell out (if there are no seats, you stand until someone gets off).
For Paris to Lauterbrunnen, look at French Rail http://www.sncf.com/en/passengers. If you can't get it work, use https://www.capitainetrain.com/en.
For Lauterbrunnen to Munich, use either Swiss Rail https://www.sbb.ch/ticketshop/b2c/adw.do?sprache=en&artikelnummer=7731 or German Rail http://reiseauskunft.bahn.de/bin/query.exe/en. If you are using German Rail, be SURE you can print the ticket at home before buying, since you can't pick it up in Germany.
From Munich to Venice, use the German Rail link above.
You print these tickets at home (they may say you need A4 paper, but 8.5 x 11 works fine).
For all of these, if you're having trouble finding the savings with advance booking, just look at a ticket for tomorrow vs. your intended travel dates. If you can live with the last minute fares, and if the route does not require reservations, you can wait. But if you want the discounts, you have to book as soon as possible; once the allotment of discounted tickets sells out, it's gone, and the price goes up. And if the route requires reservations (such as Paris to the Swiss border), book ahead.