Depends on what I'm buying. For certain personal accessories I look to Alfred Dunhill, I suppose because my late father always did. Not a department store, per the OP, rather a specialty. I used to buy at Mark Cross for my leather goods, but they went out of business, so Dunhill covers me there now, as well.
For my suits and dress shirts I go to private tailors, but for dress shoes I like Allen Edmonds, that are sold a little more widely. They use an "American" last that fits my foot perfectly. For athletic footwear I like New Balance, because they also use an American last, and have wide sizes, which many others don't, and I usually get them at Footlocker.
For more mundane things, like underwear, socks, small appliances and household stuff, I shop at Target. Though sometimes a good sale at Macy's or Dillard's will undercut even Target. I bought my last pair of Timberland moccasins (I've been wearing moccasins as my daily shoes since I was 8 ) at Macy's for almost nothing during a sale.
I also buy my partner kitchen things from Williams Sonoma because he likes the speciality items they carry. (For a Christmas gift I got him a 9-speed digital KitchenAid hand mixer at half price on a 1-day sale there -- I'm not a fool when it comes to finding bargains)
So what department store would I be? None. I shop all over. Show me the mark-down, and I'll show you my card.