Hi,
Lived in Paris for three years and visit regularly every year at least once but mostly twice a year...
Cant say much about the cheapest way to get there but Beauvais is not close to Paris... about an hour bussdrive and can be tough getting to and from all depending on when you arrive and leave... i suggest you take the eurostar from London... very smooth ride and you end up at Gare du Nord which are pretty central in Paris...
Talking about central of Paris is impossible since the whole town more or less is central... Paris is divided into 20 arrondisement (quartiers) and the number 1 is at Louvren and then it goes like a spiral around paris across the river so north of 1 is 9 if you get my drift... so basically arrodisement 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 is pretty in the middle of that spiral... so if you want to stay central i would stick to arr 1-6, 9 and 10 for Hotels... arrodisement 5 is known as Latin Quarters... pretty pitoresque and very touristic but still ok prices... every arrondisement used to have its workmans shops so for example you will find many shoestores in one area and many sexshops in one area and so on...
For nice outings it all depends on wallet but you will find Marais quarter (arr 3 and half 4) full of small resturants and boutiques, the road on the map to follow is Rue Rivoli.. to see where im aiming at is between Les halles and Batille... quarter 6, 7, 15 are Boulevard St Germain... very nice and chique areas and still very pleasent to walk around and do shopping... lots of restos with long history and great food, if you like jazz you will find most clubs around this area... resto to visit is L'entrecote... they only serve one dish... guess what... and with a herbs sauce that noone knows whats in it but it is DELICIOUS!!!!
Sacre Ceur and Montmartre is in arr 18 (northern Paris) splendid view over Paris and lots of small restos with nice prices... one resto you must visist is Chartier at rue faubourg Montmartre.. a restorant that has been located on the same spot since 1890 something and not much has changed during that time... mostly male servituers and fast service... used to be very cheap but you will only get what you order so dont expect anything more... so if you want sallad you have to order that to your meal... if you want potatoes to you meal you have to order that... and so on... well i could go on forever but if you have any more Questions just ask... ill try to answer as well as i can