If you only need an apartment to sleep in then there are very cheap rates available even in the good areas. We rented a studio apartment which slept up to 4 people easily for about £500 a week right in the centre of the first arrondisment. Some of them also do weekend/3 day renting. Look on vrbo for ideas.
As for places to go, the world is your oyster. I am a huge fan of the Hemingway Bar at the Ritz and we tend to gravitate there whenever we are in Paris. There's also a lovely bar at George V.
At the other end of the scale, Au Trappiste Bar on Rue St. Denis, near the Chatelet metro station, is also fabulous and you get all sorts in there. It's also open 24 hours which is a bonus.
For shopping I highly recommend going around the depot vente (second hand) stores such as Reciproque. You can find brand new and nearly new designer stuff, though you're still looking at several hundred pounds for a jacket. It is great fun to look at the stuff though, even if it's not in your price range. I have scored a couple of absolute bargains there. There's another great depot vente on the Left Bank but I have forgotten the name of it - it's the one run by the eccentric old lady with the pink hair, the eternal cigarette and the small dog if any Paris experts are able to help?
Galaries Lafayette is also good for browsing, don't forget to look up when you're in the cosmetic area and be amazed at the fabulous roof.
Mostly my favourite thing to do in Paris is to sit outside a cafe, enjoying a small coffee, then we wander down the street and find another pavement cafe to enjoy, do a bit of people watching, then on to another pavement cafe, more people watching and so on. Coffee will cost more if you sit outside at the pavement, less if you drink it at the bar. Paris is expensive - you can keep costs down by getting lunch from a street vendor.
You can buy a carnet of tickets for use on the metro. I'm not sure of the prices at the moment, but it is very reasonable.
If it's your first visit, concentrate on the touristy bits, so do the Eiffel Tower, a boat trip down the Seine, a walk in the Tuilleries, etc.
One other tip - take a handbag that you can wear across your body and that zips up. Pick pocketing is no worse than any other European capital but you don't want to have your holiday ruined by opportunistic thieves.
If you want to feel less like a tourist, wear stylish flat shoes or boots, not trainers.
Also, the French are incredibly polite, so make sure at least one of you knows how to say good day, good bye, please and thank you in French and use them frequently. Contrary to popular belief the French are actually very welcoming if you cover the basic courtesies.
Sorry - that turned into a bit of a missive but I hope it helps. You will have a fabulous time and no doubt fall in love with this amazing place as much as I have.