train tickets - buy in advance or at train station?

As long as you don't mind paying full price, buy your tickets at the train station. However, you can save big bucks with advance purchase, although you will be committing to a specific train.

The danger of buying in advance when you land at CDG is if your plane is late or you encounter a long delay going through immigration and customs and miss your train, your advance purchase discount ticket becomes worthless and you'll need to buy another full fare ticket.
Vernon to Paris is a regional train so can be bought at the station. There is no advantage to buying in advance.

I don't know about the first trip but the trains to Vernon are regional trains with no benefit of buying early -- in fact you can pick up a ticket a few days ahead and use it any time. I recommend that as if you don't have a Chip and pin credit card you will have to buy at the manned counter and at St. Lazare where these trains depart, that can take over an hour (we missed the train to Vernon and went the next day because of the long line). So pick the ticket up a day ahead at any SNCF boutique or train station.

For the train after arrival, it is risky to buy ahead because late planes are common. If you do this, I'd leave 4 hours from ETA to the train. It often takes about 90 minutes to clear immigration and customs and collect bags (can be less, can be more, but 90 minutes is not unusual)

Often the best strategy is, Buy in advance at the train station.

If you are arriving in a town train, you can do this on arrival, for the trip to your next destination. Since you will be at a train station anyway. Or at any time that you are certain of your next departure, at any train station or SNCF boutique

This saves you last-minute worries, but also avoids leaving you stuck with tickets if your plans change. (That's why you would prefer not to buy way in advance, right?)

You might do this for your Vernon trip. Probably not for your Amboise trip, if I understand you correctly.

At the end of our 4-day stay in Paris, we will be taking the train from Paris Gare de Lyon to Basel, Switzerland, then on to Luzern. I am somewhat hesitant about buying tickets online well in advance because there doesn't seem to be any rhyme or reason as to how to do it online, the websites are all different, and in addition some of the available sites do not allow online printing of the tickets themselves. Certainly, there appear to be savings in purchasing them from the US in advance, but I have no idea as to whether they're legitimate or not.

We will have time one or two days before our train trip to take the Paris metro to the station and buy our tickets. Are there savings then, as opposed to purchasing them on the day of travel? We will be dealing with the same issues in Luzern for our subsequent trip to Milan, then Florence.

Thanks in advance!

I also recommend that rwalker purchase tickets in France.

egbdf13579's situation is entirely different, so not a good fit with this thread, and it would be a pretty reliable savings to buy the France-Switzerland TGV ticket online with and advance discount. The amount of savings and recommended method to buy somewhat depends on when you are traveling and what fares are available. The easiest online shopping, usually for e-tickets, is at https://www.capitainetrain.com/search/ or http://ricksteves.raileurope.com/us/rail/point_to_point/triprequest.htm (which may not offer the lowest fare category). Buy whichever is cheapest.

Tickets from Switzerland to Italy are sold at www.sbb.ch; tickets within Italy are sold at www.trenitalia.com; both offer some advance purchase for specific times and both are also sold at http://ricksteves.raileurope.com/us/rail/point_to_point/triprequest.htm and in any train station in Europe. Italian tickets are pretty affordable to start with; see regular fares on the maps at http://www.ricksteves.com/travel-tips/transportation/trains/cost-maps.