Love Paris!
Walk, subway, RER. Buses if you can figure them out, but I was never there long enough to get a feel for the bus system. If you use the subway or RER, hold on to your ticket; like the Tube, you need them to not only get into the station but also out of it.
Pick ONE thing to do each day: the Louvre, the Eiffel Tower, etc. Don't try to cram a bunch. Honestly, it would take at least three days to do the entire Louvre anyway. If it's something you want to do, go one day, pick a wing (for the record, the Mona Lisa is overrated from a tourist-standpoint... your kids won't likely be able to get close enough to see it) and then spend the afternoon/evening wandering the Tuileires in front. People watch, read, enjoy a café... be laid back, like Parisians.
I would recommend doing 2-3 expensive touristy things (Eiffel Tower and Louvre? Musee de l'Armee? Musee d'Orsay?) and then make an effort to see/do free things, like walk the banks of the Seine.
My favorite thing last time I was in the city was going on free or self-guided tours of all the beautiful cathedrals: Notre Dame, Madeleine Cathedral, Sacre Couer (although Montmartre is turning into a bit of a tourist trap and you really have to watch out for pickpockets. I had a guy run up to me, try to put a bracelet on my wrist, and demand two euro for it...). We were able to do all three in one day, but that's ALL we did that day; if you like architecture and/or religious and culture history, it's really cool.
If you're going to eat in restaurants, do it at lunch; do street vendors or bread, cheese, fruit, wine, etc. for dinner. Paris has the best food in the world, but they know it and charge and arm and a leg for it at dinner time. You can get great food and sometimes even a reasonable prix-fixe at lunch. No one will bat an eye if you take an hour and a half to enjoy your meal.