I was on holiday in Paris 3 weeks ago.
Catacombs - an interesting visit but get there by 08:30. By 09:30 the queue will be round the block to where it starts and although it's not long it will take up to 4 hours to get in from there. Buy tickets there.
Eiffel Tower - buy tickets in advance, online, to the top. The queue for ticket holders isn't massive (ours were at 09:00) but do go straight to the top. You can work your way down for free but if you go from bottom to top you need to get another ticket to go from lvl 2 to the top. Visit at night too.
Notre Dame - Longish queue but it moves fast as it's free entry. To go up the towers, queue down the road to the left, as you look at the front. This queue might be long.
Louvre - Get tickets in advance as the queue will be long. You can leave and go back in as many times as you like on the same ticket but only for that day. The Mona Lisa is smaller than you think and getting to it is hard assuming you can see it past all the cameras and iPads! The Venus de Milo is better for that.
Boat tour - take one from the river by the Eiffel Tower. It lasts about an hour and is pretty good.
You can get around on foot but the Metro is dead easy to use. Tickets we used are 1.70 euros and you can go anywhere until you leave the last station. New trip - new ticket. You can get books of ten tickets but we didn't bother.
Scam artists (typically women) will try to get you to sign a petition. Dont. Tell them to go away. We only saw these people at the Eiffel Tower.
Groups will offer to put a string or braided bracelet or ring on you. Dont let them. I think the scam is for them to then march you, in their group, to the nearest cash point...
If you see a ring/piece of jewellery on the floor leave it there.
The typical pick-pockets are also about so take the usual precautions.
Scammers will typically ask if you speak English first (people are more likely to speak English even if it's not their native tongue) which helps give them away. Just keep walking. Their no bother really, as long as you don't pay them any attention.
This tourist info office was really helpful and is where we got our Louvre and boat trip tickets.
As others have said, the architecture is amazing so just wandering about can get you to some lovely places. The wide streets and parks really make the place.
We were too tired to do a lot in the evenings but there are loads of restaurants and cafes all over the place.
Basically: prepare to queue (take a brolly), food is expensive, the sights are great, the locals are nice and happy to help or give advice. 9/10 would visit again.