To be honest, there are a lot more to do in the city - you shouldn't be limiting your journey with only these following, but the ones here are the ones I recommend MOST. You can find very brief info next to every place written here, but I will write more detailed posts about most of them in the meantime.

Every destination has useful details like how to get there and opening hours and days - but I included only the métro lines that you can use as this is the best way to travel around the city (you will see the busses are all empty anyway!). If you are looking for any other means of transportation, then I would recommend you to check the web sites I wrote under every destination.

Eiffel Tower from my camera

Eiffel Tower:

The masterpiece of Gustave Eiffel - known by almost everyone - the signature of Paris (and even of France) - been the world's tallest building before it was eclipsed by Manhattan's Crysler Building - worth seeing - worth going up (you can see one of the best views of Paris from the toppest point) - the MUST-SEE piece of Paris... The legend says that noone believed it would be popular at all, before it was built, but there it is! Now one of the most popular buildings in the world. If you ask me, I would say it is mostly because its design is unique (considering the building itself is actually useless without its restaurant on the roof floor, and the design is the only thing that makes it so famous) - I should warn you about the long queues you may have to wait at some times. I had to wait for 2 hours to get in to the tower!!! But was definitely worth it.
The line is for the lifts by the way, so if you believe you can get to the top by the stairways, then good luck! Some people were doing it when I was there, but not sure if some one can easily climb them up!

Hours: Open every single day of the year. 9AM-midnight from 15th of June to 1st of September / 9.30 AM-11PM during the rest of the year.
Tickets: Depends on where you want to go; to the first floor, second floor or to the top. To check online: http://ticket.toureiffel.fr/index-css5-sete-pg1-lgen.html

How to get there by metro: 
Line 6, Bir-Hakeim - if you take this line which runs above ground, you'll get a view of the tower as you approach the station. Was an exciting moment indeed!
Line 9, Trocadéro

Web site: http://www.tour-eiffel.fr/

the view from the top of Eiffel

Museé du Louvre:

The museum was a royal residence built in 12th century, so has most of the beautiful sides of that time's art. Buildings in Paris look so similar most of the time, all Gothic or Art Noueveau or Baroque... The main facade looks magical, especially for those who love very detailed designs like I do, and the famous Louvre Pyramid is an interesting contrast of the main building. There you can find some of the unique pieces of art - like Mona Lisa, as most of you might already know. I was actually quite disappointed when I first saw the Mona Lisa portrait, which seemed smaller than I always imagined - but every one in the museum was so keen to see the masterpiece that I had to open some little space for myself to get closer to the portrait (But I've seen it in the end!) I'd definitely recommend you to bring a book about the traditional art (like a guide that tells you most of the art movements and the most popular pieces of Louvre), since there will be loads loads of famous paintings that you might regret if you don't see later on.
And the good news is, Institutions like the Louvre, Musée d’Orsay and Musée Rodin are free every first Sunday of the month! I preferred seeing Louvre on a first Saturday - so I was able to enter the museum for FREE. 

Web site: http://www.louvre.fr/

Museé d'Orsay:

I like this one as it is the home of France's national collection (I like the national collections, there I can find everything I'm looking for!) - pieces from the Impressionist, Postimpressionist, and Art Nouveau movements - Van Goghs, Manets and Cézannes... The greatest pieces althogether - aaand the building itself is an Art Nouveau showpiece, worth seeing. Never say "I've been to Paris, but haven't seen the Museé d'Orsay...
Hours: Open every day except Monday - 9 AM-6 PM in the summer, 10 AM-6PM in the winter.
Tickets:  €7; the fee is €5 for 18–25 year-olds. Free entry for children under the afe of 18. Entrance free the first Sunday of the month.
A good web page: http://goparis.about.com/od/sightsattractions/tp/OrsayVisTips.htm
How to get there by métro: Line 12, Solferino 

Web site: http://www.musee-orsay.fr/

Cathédrale de Notre Dame de Paris:

the Notre Dame Cathedral from my camera

Notre Dame, meaning Our Woman... I could go there again and again and again without any symptom of getting bored! The fame it has comes from the story of the Hunchback of Notre Dame, which is one of the most popular works of Victor Hugo - and before that, no one knew about the cathedral at all. When you look on the map of Paris, you will see the cathedral is located on a little island (kind of) and the island is connected to the other parts of the city by various different bridges.
The cathedral was beyond the words - I can only say it is magnificent! The Gothic facade and interior design had captured me - I couldn't help myself taking countless pictures of the cathedral. And it definitely is much much bigger than it looks from the outside (and on the photos). Inside, there was a service going on, with music on the background. I will give you more details about the cathedral in another post, so now I'm not going to make it long.

the service

To get there by métro, you can take the following lines:

Line 4, Station Cité or Saint-Michel 
Lines 1, 11, Station Hôtel de Ville 
Line 10, Station Maubert-Mutualité or Cluny – La Sorbonne 
Lines 7, 11 and 14, Station Châtelet

Web site: http://www.notredamedeparis.fr/-English-