Last Supper Milan and Train to Venice

You'll need separate tickets for the two trains between Venice and Padova and Padova and Venice. Look up the train options - some may be regional trains (same price at last minute as in advance), others may be fast trains that are usually cheaper if bought in advance than at the last minute (for sure Venice to Florence and Florence to Rome) - you might want to buy those tickets ASAP to get a good price. Between Venice and Rome you can take either Trenitalia trains or Italo (different company).

If there are regional train options between Milan, Padova, and Venice, you could use those and book tickets at the last minute and not worry about higher costs or trains selling out (unreserved regional trains can't sell out, though they can get crowded). If you will have a smart phone, you can try apps like Trenit to buy tickets with your phone and not even need to print them (show them to a conductor on your phone).

Here are some useful websites to get train info:

www.trenitalia.com/tcom-en (Trenitalia only)
www.trainline.eu (both)

but while planning this trip I've been told that I'd be better off buying point to point tickets a couple of days before my date of travel, i.e. buy the Venice ticket when I arrive in MIlan.

Yes and no. It is cheaper to buy tickets in Italy than it is to buy a pass, but it is even cheaper if you buy tickets for long distance trains in advance.
You can lookup train times and prices and buy "print-at-home" tickets on the Italian Railways website: http://www.trenitalia.com/tcom-en
Compare prices for tomorrow with those for May, and see the difference.

When you buy tickets for long distance trains they will be for a specific train, and will not be valid on any other train. They will include a reserved seat, and it will give your seat number on the ticket. If you want to stop off in Padua, you need to buy two tickets, one "Milano Centrale" to "Padova" (you must use the correct Italian spelling for the station names), and another ticket on a later train from "Padova" to "Venezia S. Lucia".
Same for the later trips to Florence (Firenze S. M. Novella) and Rome (Roma Termini).

Best is to buy the ticket from Milan to Padua in advance to get a possibly considerable discount. From Padua to Venice you can take a regional train (Regionale or Regionale Espresso) for a few euros. These trains take hardly any longer than a fast train (Intercity or Frecce) on this short stretch, don't require reservation and have a fixed price. Buy that ticket at Padua station, either at the window or from the machine. Those Regionale tickets must be validated (time-stamped) before boarding the train at green/grey machines in the hall or on the platform. Not doing so can result in a heavy fine.
I assume you have luggage with you. Check whether Padua station has any facilities to store them during your visit. This question has been asked on this forum, but I can't remember the reply.

Last Supper--- I would bring a copy of EVERYTHING, you got. When we went another member of our party had made the reservation. When we went to check in they " see nothing here for you". I had kept a hard copy of the email I had gotten that showed the email confirmation he had gotten from them. Without it we would have been out of luck. ( Yes, we got in)

The Scrovegni Chapel in Padua requires a time-specific reservation. The reservation must be made at least 24 hours in advance. I spent several days in Padua, so I didn't have to go through the online process. You may need to notify your credit card issuer that you're going to charge something in Italy. If the website won't take your card, try using PayPal if it is an option; that is often a good solution. The groups permitted into the chapel are very small, and you have a limited time there to protect the frescoes. I have not heard that reservations have to be made way ahead of time, but once you are absolutely sure of your timing, I'd go ahead and make the reservation. You do have to spend some time in a controlled environment before you go into the chapel, so be sure you understand how that works and allow sufficient time to get to the chapel from the train station. The luggage-storage process may take some time; it's a staffed operation.

Google led me to confirmation that there was luggage storage available at the Padua rail station as recently as last fall, so that's good news. Here's the Fodor's forum thread where the subject was discussed.

There are many interesting sights in Padua beyond the Chapel. It's worth a nice, long stop.

Agreed about Padova; I spent a night and enjoyed it. I actually preferred the Scrovegni Chapel to the Sistine, maybe because of the calmer viewing environment.

"I actually preferred the Scrovegni Chapel to the Sistine, maybe because of the calmer viewing environment. "

Not only is the viewing environment 1000 times more pleasant (since you're in a very small group, and you haven't fought throngs to get there, and everyone isn't tense), but you're much closer to the art, so you can see everything.

As said above, before you enter the Scrovegni Chapel, you watch a video while you're being de-humidified. This means that you know what you are looking at, so you can maximize your 15-20 minute viewing time.

To all this, add the fact that the Scrovegni Chapel looks like it was painted last week, and you have a fantastic experience all around.

Ditto for the Scrovegni Chapel in the late 90's, although they did charge a small fee to go in. The frescoes were dark from hundreds of years of candle light. They have since been restored to close to what is believed to be original condition, and the building sealed and climate controlled with an airlock room to stabilize humidity, and of course, timed entry with only 15 minutes of viewing time.

Yes about the need for tix to the Scrovegni Chapel. Very tightly regulated. The good news is that it is VERY NEAR the train station. And, it is an easy walk into the heart of town, the farmers market and restaurants. We spent about four hours in Padua on our way from Venice to Verona; it was delightful.

A warning ... The train conductor was very officious. It you do not follow all the rules, and have your STAMPED/VALIDATED ticket in hand, you get grief. We got a raft of it ... and it was because the validator in the station at Padova (for our second leg, Padova/Verona ticket) was broken. On board the train, the conductor scolded us for not going to him about that as soon as we got on the train. Then he went to a mixed race family of four from Australia and treated them as thieves and talked about turning them over to the police.

Fred,
Seattle