You might want to start your tour at Notre Dame. It's not a hotel, but a monument built by Louis XIV as a hospital for veterans. Go during the week when it's less crowded. The Metro subway is the best way to travel in Paris. It's easy to figure out, and if you buy tickets in multiples, you save.
Taxis are expensive.
The with-it place to be in Paris is the Left Bank. You can rent a whole suite on the Left Bank for the price you'd pay for a single room on the Right. One of the most charming hotels in the area is the Lutetia. An art deco gem, the hotel is situated on what used to be the vineyards of Paris. The staff makes you feel at home when you first step in the front door and everybody speaks English.