As has been said, it's best to skip indoor visits as it takes the longest time to get into any of the museums and you don't have much time.
Now, if the weather allows it, there's a pretty good itinerary you could take that shows you around many of the most popular monuments of the city. That's if you don't mind walking a lot. Otherwise you can do just the parts you're most interested in and explore the neighborhood.
It starts at the Eiffel Tower, then goes on to the Arc de Triomphe which is nearby (you can also take the subway).
From there, it's a straight line to the Concorde and you get to walk the Champs Elysées. You might want to take the subway again for one station to skip part of it. Then you arrive at the Tuileries, where you can start following the Seine as you pass by some other famous landmarks up to the Louvre.
You could get as far as Notre-Dame and St-Michel, but you would have to walk fast. If you prefer to take it easy, there's a nice park at the Tuileries where you can take a rest.
If you look at a map, you'll see it's almost a straight line.
Some things to be aware of: since this is all very touristy, it is hard to find a good restaurant along the way (or it won't be cheap), so I would just grab some sandwiches. Also, it can be hard to find bathrooms in the city so if you do find one, don't waste the opportunity! And finally, I highly recommend buying some subway tickets in advance. They're not very expensive if you stay inside the city and will always be handy.