As a tour guide in Paris, here's my time to shine and help you save time and money:
Avoid Paris In June and July, August. September is best, May comes in second.
ALWAYS, always book your skip-the-line tickets online before going to a landmark. Nothing in Paris is worth waiting 3 hours in line. Instead, go drink some wine and eat some cheese + baguette in a nice park or by the river.

Louvre: Mona Lisa isn't worth waiting hours and is only one of the several Da Vinci we have. Currently and until december 2019, the Mona Lisa is in a small room with up to 100 min queueing time to see it. There are 34999 other pieces of art on display in the Louvre, which has more than 10 miles of tunnels and covers 9000 years of human art.
Also, if it's your first time in the Louvre, get yourself a guide or get ready to lose yourself in the museum and be frustrated. But it's also part of the fun, I guess. Other option would be to get the Nintendo DS interactive map + audio guide.

Orsay: it's awesome, but not worth standing in line for up to 90 min in June. Buy the skip the line. Better with a guide but not as critical as in the Louvre. Audio guide available.

Catacombs: BUY THE SKIP THE LINE! I adore the catacombs tour - in fact it's my favorite to give - but nothing down there is worth waiting 2 hours in line. Also, if you get a guide, you get access to a VIP section with cool carvings and a chapel. Audio guide is option #2

Invalids + war museum: only worth it if you love Napoleon stuff and/or War history. There's a lot of that. No need to buy skip the line as the museum is almost always empty.

Notre-Dame: 😢

Sainte-Chapelle: Absolutely worth it but the lines can be huge in June: BUY THE SKIP THE LINE. Most beautiful stained glass windows in Paris.

Saint Germain des Prés: totally underrated roman/gothic church which is almost entirely restored. Incredible colors and will give you a good idea of what churches looked like in the Middle Ages.

Versailles: BUY YOUR SKIP THE LINES. Seriously. Worth it, but show up at the earliest in the morning if you don't want to be swimming through a sea of tourists as you pass the royal apartments. I don't even do this tour since the conditions are so bad.

Eiffel Tower: For the ET, it's easy: either you've booked your skip the line months in advance, or you're screwed. The lines can get absurdly long and again, it's not worth your time. If you don't have a ticket but still want to enjoy the view of Paris, go up the Montparnasse tower (no need to book in advance) or up the Arc of Triomphe (cheaper and better view, if not as high IMHO). Arc de triomphe skip the lines can be booked online. Don't eat at the restaurants up there. You will pay for the view, not for the food.

Boat cruises: Yes, worth it, but BOOK ONLINE (you're starting to get the point, I hope). Get the Vedettes du pont neuf. Their boats are smaller and you don't feel like an ant. All the companies show you the same stuff and the cruises are all one hour long. DO NOT book a dinner on the river. the food is shitty. If you do want a decent dinner on the river, you'll have to go to an expensive company like Yacht de Paris. But they're pricy.

Since this post is attracting attention:
MORE STUFF: There will be a huge Da Vinci exhibition at the Louvre from Oct 19 to Dec 19. It's going to be great but also a fucking mess, as usual in the Louvre. Don't expect to see it if you haven't booked right now. I don't even know if there are tickets left.
There is also a super cool and trippy Van Gogh exhibition at L'Atelier Lumière until Dec 19 (The Klimt one will also be back on Aug and Nov 19).
Just as amazing, is an exhibition about Naïve art (Le Douanier Rousseau, yay!) at Musée Maillol from Sep 19 to Jan 20.

Also, if you are travelling with kids from 4 to 12 and the weather is good, take them to the Jardin d'Acclimatation. They've renovated it and it's even more awesome than when I was a kid. It's like a mini theme park.
Another option is the Cité des Sciences et de l'Industrie. Think science museum for kids with lots of hands on stuff like the Exploratorium in SF. Really fun and right near a huge park with playgrounds, a fucking submarine you can visit (and drinks for the parents).

EVEN MORE STUFF: Most churches in Paris are FREE. St Eustache, St Germain des Prés, St Séverin, Sacré Coeur, St Pierre de Montmartre... For Sainte-Chapelle you do have to pay 10€ since it's not an active Parish but it's 150% worth it.

! Important !: If you plan on visiting many museums and landmarks, buy yourself a museum pass. It will save you money and skip almost all lines. Go to their website and figure out if it will save you money.
Same is true for the Metro: get yourself a metro card (Paris visit pass) and save money of the uber/taxis.

Last but not least: I have spent the last 5 years working on a map of my favorite places to go out, eat and drink in Paris. Here it is free of charge, just because I like you Reddit :)

Check out this website (partially in EN, mostly in FR) for what to do in Paris.

AMA