You are spoilt for choice restaurant wise.
Favourites for us are Baracuda on Rue Meybeyer. Just up from the front and one block along from the Negresco. The staff are really nice. They also have Koudou on the front and for a special treat the tasting menu there is amazing.
Booking in advance is needed for Oliviera (it's on rue du Collet in the old town). It's run by a husband and wife and they make their own olive oil and each course is designed around an accompanying oil, even if you don't eat there it's worth buying a bottle to take home, it's closed Monday's and Tuesday's. My favourite ever in Nice.
La Trattoria on the Rue de France is great for a pizza or pasta. It's also near the Negresco. We tend to avoid the other end of the Rue de France and the Cours Saleya as they are busy, not as good and more expensive.
We tried a lovely place this year which was on the street that runs up the side of the meridien. I think it was called La Massarena? It was on the left as your walked up from the sea.
A bus anywhere is €1.50 and you can pick up a 81 or 100 from the top end of the park which the fountains are in. It will take you to Villefranche which is a lovely village with a sandier beach than Nice. You can also get the train. It's much easier to get in and out the sea here. It can be quite choppy at Nice and the stones can be steep depending on the sea.
St Paul de Vence is worth a day trip depending who long you are there for. It's a gorgeous see town full of art galleries. I think it's the 400 bus to here but not sure.
Eze is nearby too and has fantastic views over the Med. You can get the train then a bus up the hill. Chateau Eze has a veranda overlooking the sea and while expensive (Michelin starred) it's nice as a special treat.
Where are you staying? It's a fab city and even if you stay within Nice itself there is lots to see. The old town is nice and shady during the day for coffee. The prom is very long and there are covered sections to stop and read your book. You can get a lift up the the old castle and the waterfall or there is a little train which takes you up.