day trips or one overnight trip ( By train) from Amsterdam

There's no need to rent a car. Train connections from Amsterdam are excellent. You can get to most of the towns in the Netherlands in less than an hour. My favorite was Haarlem because of the Corrie Ten Boom museum - it's a nice compliment to the Anne Frank House. You can also reach Antwerp, Belgium in an hour. I haven't been there, but I think it sounds like an interesting destination.

If you want to go overnight, you could go to Brussels. I love Brussels, but I know not everyone does. You could perhaps combine Brussels with Bruges or Ghent. It would be about two hours to Brussels by train, and then 2.5 hours back to Amsterdam from Ghent, or 3 hours back to Bruges.

Brussels is very much a hate it or love it destination. Antwerpen, Gent and Brugge are more immediately appealing. Gent is a good base: 30 mins from both Brugge and Brussels, and 1 hr from Antwerpen, all on trains which are running several times per hour.

Closer to Amsterdam, there are some lovely small towns in the Netherlands. I visited Delft a few years ago and loved it - charming and small, very quiet compared to Amsterdam, easy to reach by train (like so many places). No point at all in renting a car with such a great train network in this part of Europe.

If you want to visit Belgium, Brussels and Antwerp are both worth visiting, but I really didn't care much for Antwerp personally. Brussels is what it is: a big city with a lovely main square (the Grand Place), but I much preferred Bruges and Ghent to anywhere else. Bruges is really touristy but i loved it anyway. Ghent is less touristy for sure and still quite charming - the two aren't identical but both have lovely canal scenery. Antwerp has some interesting museums but I found the charm lacking - a couple of nice squares and an amazingly beautiful (big) train station, but it's not a place I'd want to return to.

I wouldn't recommend going to Belgium when there are many delightful places in the Netherlands that are easily accessible from Amsterdam.

Of the places I've been, I'd recommend Delft for an overnight. From Delft you are a short hop from Rotterdam and from The Hague, so you could easily explore one, two, or all three of these during an overnight visit. Delft is delightfully charming. The Hague has the wonderful Mauritshuis, a small but fabulous art museum. And Rotterdam is a real contrast to other places in the Netherlands, as it was almost completely rebuilt after WWII and has some fascinating modern architecture. I got a map from the TI at the train station and did a great walking tour, and stopped at the excellent Museum Boijmans Van Beuningen.

For day trips, Haarlem is close, easy to get to, and filled with great sights.

And I keep hearing great things about Kinderdijk, which I didn't fit in to my trip.

A much overlooked, but charming destination is Enkhuizen, only 1 hr from Amsterdam on a half-hourly train.
Kinderdijk is indeed not to be missed. See this post with practical information on how to reach it from Rotterdam and to combine it with wonderful Dordrecht.
Den Haag (The Hague) and Delft are easily done as a combined day trip: only some 15 minutes from each other with frequent trains.

We went to Kinderdijk two days ago and thought it was wonderful. Today on our last day in the Netherlands we took a day trip to Hoorn & also Enkhuizen for the open air museum there. It was terrific! Huge thanks to TonfromLeiden for telling us about those two great spots. The museum was so well done & Hoorn was truly a gem of a town.

Not only the open-air museum in Enkhuizen is worth a visit, but the whole town is a beauty. It has a village feel, notwithstanding its rich past.

Just returned from two weeks in Holland and Belgium.

Enkhuizen is fun and easily reached by train. Might also try Haarlem.

We stayed in Zandaam and rented a house. It is about 20 minutes by train into Amsterdam. Charming town and it is near Zaanse Shanse, another living village worth visiting.
Since your time is so limited, I would just stay in the Netherlands and use the trains - a great way to get around!