What to do with 5 days in Nice?

I definitely recommend a day trip to Antibes. The Picasso Museum is a treat, as is the town itself. Also, Villefranche-sur-mer, in the opposite direction. I also recommend getting a Pass SudAzur Explore. It's ideal for exploring the entire Alpes-Maritimes region (including buses, trams, and regional TER trains). A 3-Day Pass is €35 and a 7-Day Pass is €50. We got the 3 day pass when we were in Nice last September and it was perfect for all our transportation needs.

I took the bus to Monte Carlo for like €2. Fabulous views. Visited the aquarium where Jacques Cousteau worked, as well as Prince Rainier's car collection. I was with a 12yoa boy, so... Didn't go to the casino, as I had already been.
In Nice itself, the Chagall Museum was great. We were there in late November and pretty much had it to ourselves. Castle Hill offered a stretch of the legs and fantastic views. I loved shopping at the market at Liberation.

We are also having six nights in Nice, also in late October, and I've just been working on our itinerary.

At this stage, I would like two days in Nice to explore the city itself and three day trips.

So far I am looking at an Uber or Bolt to Eze Village because I love views. Then we will hopefully Uber or Bolt to Monaco and explore the town and the Princes Palace. A quick casino visit for photos, but no more because we won't be dressed for it.

On one of our in Nice days, we will train back to Monaco so that we can enter the casino for my husband to have a little gamble. I will watch as I do in every casino we've ever set foot in.

Another day trip will be to Grasse, as the home of perfume sounds interesting and it will be nice to get away from the coast and see some countryside. We may or may not do a tour of one of the perfume houses, more research to be done on that.

Our final day trip will be to Cannes and Antibes.

I know we are missing lots and lots of small towns, but you can't do everything in five days.

Hopefully this gives you some ideas, and we will be buying the Sud Azur pass. Whether it is financially sensible remains to be seen, but it is easy and that is what holidays are about.

Where are you staying?

We are staying at the Hotel Apollinaire. It is close to the Durandy station which will make it easy for us to catch the tram to the airport the morning we leave. It is also located in a good spot for us to take an architecture walking tour of Vieux Nice with Anthemion Tours. We love architecture and we love walking tours. The owner operated a similar company in New Orleans before she moved to France and she was fantastic.

Monaco and St. Paul de Vence

A couple of other villages we like: Tourettes-sur-Loup, Biot, Vence, St Paul de Vence. Absolutely try to see the Villa Ephrussi de Rothschild in St Jean Cap Ferrat, for its own beauty and for the views of the Riviera coast. And we really liked (much to my surprise) St Tropez, which has beautiful little streets for strolling away from the posh port.

As for Monaco, opinions seem to be divided into two camps. I guess I come down in the middle. If you must, go take a look at the casino (where you’ll mostly see tourists like yourselves, rather than movie stars) and see the palace area - but the rest is just apartments for billionaires. Once was enough for me.

Lots of great responses here. I'll add a couple of ideas also.

1. Matisse Museum and the adjacent Roman ruins in the Cimiez district.

2. The little town of Menton on the Italian border. You could even consider going a little further into Liguria.

I don't believe I've ever had food as delicious as I had in Nice and Villefranche-sur-Mer. The razor clams are amazing, as is all the seafood. La Mère Germaine in Villefranche-sur-Mer is truly worth a splurge IMO; grab a table by the water and put yourself in the hands of the maitre d'hotel.

We were recently in Nice for a few days and loved it! One of the favorite things we did was take the Nice Food and Wine Tour with Nadia, who Rick Steves personally recommends. She does a 4-5 Hour walking tour, giving you Nice’s history as well as an amazing culinary experience. She is excellent and it was a five star tour. Olives, cheese, olive oil, multiple wines, socca flatbread (chickpea), charcuterie items, wandering the markets pointing out all the fresh fruits/vegetables/flowers, amazing chocolates and other candied fruits, beef stew, vegetables, many types of breads; go hungry !
https://www.nicefoodandwinetours.com/
We also did a 1+ hour Mediterranean cruise that we found on on Viator, leaving from the port. It was nice to experience the area from the water. There was inside as well as outside seating, commentary in both French and English.
If you go to Eze, go early. It gets crazy busy after 11am. Very touristy and the streets are lined with shops. Amazing views.
We had one of our most memorable meals in France at Hotel Le Negresco, not in the fancy dining room but at La Rotonde. We dressed up a bit (skirt and nice top for me, Oxford shirt and nice khakis for hubby) and walked into the Hotel (which is an art museum as well, but access only to hotel guests) asking if we could get a reservation. I had tried online and nothing was available. They let us right in and we had our choice of tables. The service was outstanding as well.
We had planned to go to Villa Euphrussi but the parking lot was full by the time we arrived and so we missed seeing the house and gardens.