Groceries: Sainsbury's, Tesco; upmarket: Waitrose, Marks & Spencer (M&S). Whole Foods is here as well. Pharmacy and "personal care" (what an odd euphemism): Boots, Superdrug.
Home Goods: Robert Dyas has everything from appliances to kitchenware - I very happily stumbled on one of these right at the start. Muji is great for storage; also has an eclectic assortment of pens, gadgets, bedding, clothes, etc.
Stationers/Office Supplies: I really like Ryman; it probably most closely approximates Staples. Paperchase is nice, and by nice I mean fancy and boutique-y.
But wait, there's more:
Budget Clothing: Pretty much all major US brands have a presence here. It's London, after all! Except for some unknown reason, TJ Maxx is called TK Maxx. I haven't really had occasion to clothes-shop but Primark, Asos, Uniqlo, and H&M seem reasonable if you're looking for budget-friendly trendy. (Topshop is essentially the upmarket version of H&M.)
Electronics: Maplin; I live across from one though I haven't needed to use it so far.
Of course, there's always Amazon.co.uk. Argos is another online store that I haven't used, though it seems reasonable in pricing/quality. I hope these can smooth your path!