Best way to cook turnips?

I agree with SFMiller,the more thinly sliced the turnip roots the better. I prefer adding one third sweet potato,turnip root,and yukon gold as my root gratin layers, in between bechamel layers. As for turnip greens, cook them with pork flavoring( ham scraps,bacon) in someone else's house-they are stinky !!

Add to mashed potatoes, or add to soups as cubes or matchsticks. I also like them raw (matchsticks again) in salads.

I like them halved/quartered/chunked and roasted with olive oil and salt and maybe rosemary or thyme. It tames the peppery-bitter edge that turnips have, especially ones that are older or were grown in hot weather. They're also nice sliced thin and baked in a gratin with cream or bechamel. If they're young and sweet, try them raw, sliced thin as a crudite or grated as a salad, dressed with a tart vinaigrette or slaw dressing of choice.

boil and mash just like mashed potatoes

Around here...they're normally cubed and braised with bacon, and chicken stock.
Or Cubed and used in winter/fall greens. Kale, turnip greens, collard, mustard greens. Cooking the greens first with pork belly cubes, rendered and then add greens and stock...and turnip cubes at the last 20 mins of cooking (out of about an 1 hour simmering time).
Make some corn bread if you do greens to soak up the 'pot liquor/likker" juice from the bowls of greens.

I never really cared for them myself...but in greens they're okay with a fine dice.