How much do you save by getting advance train tickets?

I know I've read it's less expensive (which of course I like), but is
it a significant savings?

Look up the prices tomorrow and compare with the prices in a few months and you get an idea.

If I do decide to go to Paris for the day, is there a particular train
station I should use, one that's closest to the tourist spots?

You don't have a choice of station in Paris. Trains from Strasbourg arrive at Gare de l'Est.

One thing you can do is see if there are discount passes available.

A lot depends on the country and the trains. Regional trains usually do not need advance booking - you pay the same amount no matter when you buy it. However, many high speed trains do have higher prices the closer you buy to your departure date, so you can save a considerable amount of money by buying your tickets in advance.

in the UK even many slower regional trains (we don't use the R designation) benefit from Advance tickets on the longer runs.

It is on the fast trains (TGV in France, Frecce trains in Italy) as well as cross-border IC and EC trains where you can save money. How much you save depends on the route and how far in advance you buy. To find that out, I suggest you do as Badger says and price it yourself, comparing the cost of a ticket purchased for travel from Venice to Florence tomorrow with one for travel in early October. Use both Trenitalia and Italo for that route.

To answer the question in your title: A few years back I saved 496 CHF by buying tickets 2 months in advance for our family of 4 to travel roundtrip Zurich to Milan and Milan back to Zurich. That was just about $500. I have been buying our tickets for the fast trains well in advance ever since.

I think I've saved 50% on RT Antwerp-Cologne. Do you really need all that "flexibility", or does it just feel good??

I also hate lining up for a ticket window, or not getting a seat.

I do most of my travel in Germany, on regional trains. There are no (that I know of) discounted, advance purchase tickets on regional trains in Germany. However, there are significant savings available by buying regional passes, Länder tickets (passes for regional trains in a German Land or state), or the QdTL (Quer-durchs-Land-Ticket) pass for all of Germany.

Discounted, advance purchase tickets are only available for a trip that includes at least one leg on a long distance train. In 2017, I took such a trip from St Goar to Bingen to Ulm to Kempten to Pfronten. The walk-up (Flexipreis) fare for tomorrow would be 104,40€/P. If you bought today a ticket for mid-September (a Saving Fare ticket with limited cancellability) the price would be 53,90€/P. That's almost a 50% savings.

I tend to use high speed trains only for longer trips, and I use the advance purchase discount when I can count on making the specific train (i.e., not on the day of arrival).

For Strasbourg-Paris and back, yes, you can save significantly by purchasing in advance. Or you can pay out the wazoo by purchasing last-minute.

A last-minute round-trip from Strasbourg to Paris won't leave much change out of $200, whereas advance tickets can be had for less than $100.
If you do wish to travel last minute in France, and if you are eligible, a Carte Avantage Senior can help: it guarantees at least 30% discount on full fares.

It's not a case of simply buying tickets in advance but also understanding the different ticket types.

The cheapest tickets often have the least flexibility; more expensive tickets have greater flexibility. Both types can be bought in advance.

I always buy my train tickets ahead of time - a few months, to obtain the savings and sometimes ride in the first class when the difference in cost is minimal. But, I don’t buy them for the arrival day since the time is locked in from an advance purchase.

Check the date two months in advance of today to see the difference in cost.

I’ve traveled a lot by train and know that I like to pick a time that’s around 8-9am to leave, so my selection is predictable and not such a risk to purchase ahead.

A few years back on the day they were first released for sale online, I bought a pair of “Prem’s” TGV train tickets from Paris to Strasbourg 🇫🇷 for €19 each. The price, if I waited to buy the tickets the day before the train departed, was more than €100 each.
That was when the Prem’s tickets were nonrefundable and unchageable. Today they ARE changeable AND refundable within certain timelines that are quite generous to customers buying them.
Since then, I’ve literally saved thousands of euros by buying high-speed European train tickets online 3-4 months In advance of the train’s departure.

The website to buy them on is www.SNCF-connect.com. For English, scroll down until you see the flags. and select the British 🇬🇧 Union Jack.. You can sign up for e-mail alerts and you’ll be notified the day your desired Prem’s tickets go on sale. Some tickets can be bought up to 9 months in advance.
Have a great Trip!

Thalys one-way premium ticket from AMS to Paris Nord is €150 if you book early; eventually it can go up to €220.

I knew I'd get good input from all of you. Thank you! It's been a busy day, but I did get a chance to input a few trips for now-ish vs October and so far it looks like you're right that I should book the trip to Paris in advance since that's a pretty significant savings.

It looks like for Milan to Venice and Venice to Florence there isn't much savings by purchasing in advance, although that's just on the Trenitalia app. I need to download the Freeciarosa (Portal Freece, right?) app and look at it.

From Zurich to Strasbourg, Strasbourg to Lucerne, Lucerne to Chur, Chur to Varenna and Varenna to Milan are regional trains as well as the day trips I'll be doing to Colmar and somewhere in Germany, maybe Freiburg, Triberg and Gengenbach (I haven't decided on that yet).

I really appreciate the help you're offering. There are so many details to consider that it can get a bit overwhelming!

for your two Italian long distance trains don't neglect Italo treno. They are the competitor on the long distance fast trains and are often less expensive. Some prefer them, including me. If you sign up to their newsletter you can usually get an additional 10 to 30 percent off. Just be prepared for several newsletters a month, and they are in Italian (google translate can be your friend).

Well I purchased my tickets. That’s not the easiest site, so goodness I hope I did it correctly. There’s only one main train station in Strasbourg, right? When I was ordering, I thought it said Gare, but the tickets just say Strasbourg, no letters, whereas the Paris end of it says EST.

Gare means station. There are multiple stations in Strasbourg but if it only says Strasbourg on the ticket that means the central station.

Thanks Badger. Once I’m there and settled a bit, I’m sure I’ll get my bearings, but I’m so glad to have this forum to help meanwhile!

you have the correct two ends - Strasbourg and Paris gare de l'EST.