You should make the most of being wherever you happen to find yourself; even if you're on a mission to tick off sites you can still take in the feeling of the city while you walk. Walk as much as possible, take side streets instead of main roads, don't take the same routes to and from somewhere.

I grabbed food when I was hungry and ate on the go, you'll never be far from someone selling crepes (banana and nutella!).

The metros want exact change.

Always have a rucksack that's large enough to carry all your stuff. Make sure you have a bottle of water, maps, guide book, sunglasses, umbrella or rain mac. Remember you'll end up buying souvenirs and you don't want to end up carrying multiple carrier bags.

Remember that between checking out of your hotel and going home you'll have to carry everything with you, this nearly killed me on my last day!

Plan routes before you leave, but don't become too reliant on them. I had whole plans that went off the rails because I got lost, couldn't cross the road, got distracted.

JUST SAY NO to anyone offering to draw your picture, tie you a friendship band... anything. Just learn to do it automatically.

If you arrive in Gard du Nord, Montmartre is a good place to start your visit as it's just a short walk and the sort of Paris you'll be expecting if you've seen Amelie with its cobbled streets and squares. It also has one of the highest points in the city you can see most of the other sites.
Sacre Coeur - climb up the steps through the gardens and look out over the city. The church itself is beautiful. There's an additional charge to go up to the top and it's 280 steps, so not for the faint of heart, but a great view from the top.
Place du Tertre - the most common presentation of Paris - the centre of the artist's quarter has street artists, mimes and a lot of tourists!
Montmartre Cemetery - the entrance is a bit hidden, make sure you use your map. It's packed solid of graves and people hunting the celebrities.
Moulin Rouge - I just stood and looked at it rather than go to a show to be honest, but it would be a good way to spend an evening if you've got the cash and time to spare.

I made the mistake of going to the Eiffel Tower early in the morning, while this meant it was nice and quiet and the queues were short, it also meant that almost the whole thing was surrounded by clouds! There's a number of options for tickets and it can be a little overwhelming, I'd recommend shelling out to take the lift up as they're an experience in themselves, but it's probably not worth stopping at multiple stages, the queues for th e lifts can be very long and the views are much the same. I was quite impressed by the information boards available, finding them unusually interesting. The cafe is also pretty good. Take the stairs down if you can, I spent ages queuing for a lift because the stairs were closed.

The Trocadero over the river from the Tower offers some good views and an interesting fountain.

In the other direction are the Champ de Mars which is a good route to take towards other sites. The peace memorial at the far end is an interesting glass construction offering some nice photo opportunities and the École Militaire is an interesting building I should imagine, but it was hidden behind a huge construction site when I went.

I recommend seeing the Arc de Triomphe at night, by day it is an impressive but cold vision and the traffic zooming round it is terrifying, by night however it is a beautiful and haunting memorial. Linger around the bottom a bit to fully appreciate the carvings and the Tomb of the Unknown Soldier. Then take a deep breath and climb up 280 steps to the top. Skip the museum and head straight outside to take in the amazing 360 degree view of the city. I wasn't overwhelmed by Paris by day to be honest, it suffered badly from smog, but by night it's beautiful.

In some ways museums and galleries are not great for whistle stop tours, you're kind of trapped indoors and feeling guilty for not lingering over each item. On the other hand they do give a good opportunity to hide from bad weather and sit down. My policy involved heading directly for items that I knew I wanted to see, glancing at other things I spotted on the way, and taking the opportunity to sit on benches and appreciate the things nearby. If you don't speak french, most of the time you won't be able to read the information anyway, so there's no need to feel guilty!
The Louvre - the biggy. The big thing here I found was that the architecture was generally more impressive than the artwork. It's an amazing building and worth walking round to see all the ceilings. I'd recommend going on one of the evenings with late opening and then emerging once it's dark outside to see the pyramid.
Pompidou Centre - take the escalator all the way to the top for another good view of the city (weather permitting). I found it very easy to get lost in, so keep checking the maps! Make sure you check out the weird and wonderful Stravinsky Fountain outside.
Musee d'Orsay - a beautiful converted railway station with an amazing central hall full of sculpture. The side rooms with paintings are less impressive, but there's a lot of familiar works in there.

There are dozens of beautiful churches in Paris. Outside of the large ones it's well worth just popping into ones you walk past on your route.
Notre Dame - if you're only going to do one church, it's got to be this one. The square in front is a great place to sit and eat lunch, there's usually some sort of music playing and you can just sit and stare at the beautiful church, the hundreds of tourists, or the sparrows living in the hedges. Make sure you walk around the back to the quieter gardens and check out the flying buttresses.
St Eustache (Les Halles) - a breathtaking, huge gothic church with amazing painting and sculptures that's a bit off the beaten track so a lot quieter than Sacre Coeur or Notre Dame.
St Germaine - one of the oldest churches (10th century) in the city, very different on the outside to the other gothic buildings.
Saint-Suplice - another familiar looking church but with an interesting fountain outside.
Sainte-Chapelle near to Notre Dame. This was my only failure on my trip, the queues were too long when I went and I couldn't get in. It's supposed to be quite amazing though and is highly recommended by others.

There's a couple of other key buildings that are worth at least walking past. It helps if you know a bit about their history, so it's worth reading up a bit in advance.
The Royal Palace - I don't know what the weird collection of columns in the courtyard was all about, but the gardens and colonnades are nicely peaceful and there are some interesting old shops nearby.
The Pantheon - a very strange place which has a giant Foucault Pendulum, modern art and catacombs all in a magnificent old church.
Les Invalides - a military hospital and veteran's home, but the big attraction is the tomb of Napoleon, an amazing domed building with a very grand interior.