Nice is a beautiful city with a very good public transit system so getting around from anywhere is not an issue. However, your flight may be outside business hours which would mean some sort of taxi service.
I was in Nice last year for 8 nights and stayed at 2 seperate hotels. Both were very good but on the pricier side, especially the second one.
Best Western Plus Hôtel Masséna Nice. The hotel was in a great location, close to everything while being in what felt like a local neighborhood. Lots of nice eateries. The hotel was clean, modern and the staff were really nice. I think it is worth the price would stay there again in a heartbeat. If this is outside your budget, look up the location and any hotels nearby would be worth it.
The hotel is a 5 minute walk from Boulangerie La Niçoise and this neighborhood is worth staying in just for this bakery. We found it randomly and visited it for breakfast every morning the entire time, even when we changed hotels and the walk there was longer. We had a breakfast every morning (they have coffee/tea and a patio outside). They also had wonderful sandwiches which we took on day trips a couple times.
The second was Le Meridien Nice. Price was probably double (Sept so it was very pricy). It was in an amazing location (probably as good as it gets in Nice). The hotel was modern with very good service but you are really paying for the location. The hotel is no where near as fancy as the price may indicate which I actually like as I find "fancy" hotels too stuck up for my liking. This is as close to Old Town as I would be willing to stay.
The third option would be Old Town but I would not want to stay there purely due to the traffic and noise. Another option is the old port and the area north of it. We spent a day in that area and it was beautiful and quiet.
If you want to be close to train stations then you need to be more north which I would not recommend.
