Rudolph, loads of hotels around the city centre. look up the usual sites. rooms.com, venere.com, ratestogo etc. i was there last march, its pretty cold at that time of yr. Hope u have spanish as the locals dont speak english execpt around the tourist areas which is not where you want to be eating. none of the places we ate had menus in english and we had to rely on pigeon spanish and pointing...good craic when u point at something on the menu and see whats comes out. If into football go see real madrid play. they sell tickets there but i would get them well before match time. i would not recommend the place we stayed as it was very pokey.
i think the taxi from the airport was reasonable but we got the train out there on the way back for some reason which was ok as long as u are not carrying loads of baggage.
shopping in madrid is not that good, not as good as barcelona but i enjoyed it just make sure u bring warm clothes