The very heart of the Latin Quarter (rue Mouffetard) will be loud and dirty throughout the evening; if your sleep is valuable, I would recommend staying off that one street.

The rest of the area is nice and centrally located. Staying towards the center of the Latin Quarter will mean that Metro stations are (relatively, for Paris) far away - 10 mins walk - but that you will have a variety of lines at your disposal. From here to the sights, the RER will be your friend: take the RER B to St Michel-Notre Dame and change to the RER C for the Eiffel Tower or Versailles, or change to the RER A at Châtelet-Les Halles for Etoile (Arc de Triomphe).

In the Latin Quarter, I've stayed at the Comfort Hotel André Latin, though admittedly this was on someone else's dime, and I wasn't convinced it was the best deal. Check the booking sights (booking.com is good for Europe) for nearby values.

I've also stayed in other, quieter areas of Paris that have better views and still didn't feel like I was away from the sights. Hotels near Etoile or Trocadero can provide stunning views of the Arc de Triomphe or the Eiffel Tower, respectively. This area also has nonstop or one-stop Metro/RER service to the main sights of Paris.

Finally, my most important advice for a first-timer to the City of Light is to relax and enjoy some time just sitting down in a café having something to drink and people-watching*. This is what French people do, and is IMHO the only way to not let the grit of Paris get on your nerves. Outside of southern Europe (e.g., Spain and Italy), Paris is possibly the only big Western city to figure this out, so enjoy it.

Peace.

* In France you're allowed to just sit at your table when your drink is done without being obliged to order anything else from the waiter, aside from your bill when you'd like it.