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It is perfectly possible to do weekend trips to Rome and Paris from Nice, but that won't be "budget". Both cities are expensive. If in addition you have to travel on weekends (Friday night or early Saturday morning and back an Sunday evening), train and air fares will be high.
For Rome, flying will be your best bet. Check the usual sites (e.g. Kayak, Matrix Airfare search, ...). For Paris, the train may be interesting too. YOu can choose between high-speed trains (TGV and IDTGV) and a direct night train. Planning in advance will help to keep costs low.
Given the distances, ride-sharing and hitchhiking are no options for the trips to Paris and Rome.
The following two discussions provide some useful information:
How to get from Nice to Monaco by public transport?
If you have only three weekends, you can easily spend them closer to Nice. There are a plenty of possibilities, Monaco being one of them ...