I really disagree with Steve that 10 days in Paris is too long, but then I am just arrived and will be here 6 weeks (time limited by my husband's eye treatments at home every 6 weeks). We spent 6 weeks last fall here and another 6 weeks in the spring. For a first trip, 10 days is perfect. Not only is Paris rich with things to do, but it is a great base for day trips. There are a couple dozen fabulous day trips to towns, chateaux etc easily reached on public transport. Here are a few and only a few we have taken:
April in Paris is lovely and if lucky you will also have good weather. This is when the chestnut trees all over Paris are in bloom with their pink and white cones and the wisteria is at its peak -- lovely time to be here.
5 nights in Nice is also great and Nice is a good base for visiting other towns along the riviera. Easy to visit Monaco etc from there. I personally would rather spend that time in the Dordogne. We have done a week there twice in small towns with a car to visit chateaux, towns, prehistoric caves. Our favorite part of France outside Paris. But Nice is also nice; 5 days is IMHO plenty.
I would add a few nights in Lyon or add the Dordogne or another French area and cut Switzerland down, but I am not a big fan of Switzerland which is very expensive and IMHO not very pleasant. But if you are doing mountain hiking then it might be a great choice. Everyone gets to do the trip of their dreams and our experiences with Switzerland may be the result of our own bad planning.
By spending large hunks of time in few places you cut down the loss of time and the aggravation of logistics of frequent moves and have a chance to really enjoy where you are.