Stay as central as possible. It can be tiring to sightseeing so you might like to go to the hotel to rest a bit, put on other shoes whatever so it is impractical to stay out of town. Also you might feel like taking a cab home if you have been out in the evening.
First thing to do is to take a sight seeing coach of Paris so you get an impression of the city. Best is this hop on and off busses
http://www.conciergerie.com/opentour.htm
Then you not only get around easely on the same time as you get information about what you see.
You can always get off and take a closer look and the get on again.
The next days you can then see the museums you wish to see.
As a single woman you can go out and eat any time - but if you feel insecure about being out in the evenings take your main meal during the day where ever you are. In the evening you can go out for something smaller around your hotel.
If you have the money there are also coach tours for night sight seeing which could be of interest for you.
Have fun and enjoy yourself.