I live near where you're staying, so here's what I'd suggest - tube/bus for getting to Selfridges and Harvey Nicks. I can't remember we Harvey Nicks is, but Selfridges is on Oxford street, about halfway between Oxford Circus and Bond Street. There are TONS of buses which go from that area to Oxford Street (normally from Theobald's Road/New Oxford Street). There's the 98, 8, 242 ... could go on.
I would recommend getting an Oyster card, and going for pay as you go. Unfortunately there aren't any things like 3 day tickets, but with PAYG, there is a daily limit, so if you do find yourself using tubes and buses a lot, it's capped.
Russell Square is also relatively close to Covent Garden; you can walk there in 10/15 minutes. Tattons Brasserie is nice for drinks/cocktails, as is Freud (on Shaftesbury Avenue), but is VERY tiny and difficult to get into.
Russell Sq. is also VERY near the British Museum, if that's your thing, and the South Bank (again, quick bus rid or 20/30 minute walk, depending on how fast you walk).
I would recommend (for the views) taking the 168, 68 or number 1 bus going to or from South Bank to Russell Square area and sitting on the top deck - you get the best views of Houses of Parliament, London Eye, St Pauls, The Gerkin, all the way to Canary Wharf as you go along Waterloo Bridge.
If you need more ideas, check out Time Out online.