Turnip greens are a very common food in Southern US cooking.
The smaller baby leaves are good for salads since they have a milder flavor and are more tender.
Larger leaves are best cooked.
You'll definitely want to rinse them thoroughly in a sink full of cold water.
They tend to have a lot of dirt and bugs on them.
Just fill the sink and dump your greens in, jostling them around with your hand.
This allows the sediment to sink to the bottom.
The typical southern preparation involves boiling them for 30 minutes, dumping the green water (and much of the bitterness), and then boiling another 15 minutes with some bacon or ham hocks.
They are then drained, salted to taste, and served with either tabasco, vinegar (malt is good), or butter.
cooking.stackexchange.com/users/60/hobodave
The flavor of turnip greens is very much like mustard greens, just with a bit of the sharp mustard flavor removed.
The grocery stores here often sell turnip greens right alongside the mustard greens - without the turnips, since presumably they're a slightly different variety, selected for leaves, not roots.
These large leaves are probably a bit tougher than the somewhat smaller ones you're likely to have on your turnips.
Incidentally, I see in On Food and Cooking that mustard greens are Brassica juncea, a cross between Brassica rapa (turnip, broccoli rabe, bok choy, napa cabbage) and Brassica nigra (black mustard) -
so the flavor similarity is perhaps to be expected!
I use them pretty much as I would mustard greens, in a variety of things - as a standalone dish, in soups, in stir fries, you name it.
Depending on your tastes, they might be too strong for salads, and they're also a bit tougher than traditional salad greens.
Try a bite and see what you think
- I expect if you use it in a salad, you'll probably want to mix it with milder things.
To find more ideas, just search for recipes mustard greens.
Radishes are actually in the same family as turnips, and just like the radish itself, the greens have a sharper, perhaps peppery flavor.
I usually just toss them in when I'm using other greens, to add a bit of zing.
Have a bite of those too
and I'm sure you'll decide that they could be an excellent addition to various dishes.
In traditional US Southern cooking, one of the more common places to find these greens, all the bitter greens get boiled forever, to make them extremely soft - but I find that I actually like them cooked more quickly, leaving a pleasant bit of texture.
I also don't think they need much special treatment to counter the bitterness (as hobodave mentions )
but I do admittedly have a higher tolerance for bitterness than many.
A couple example dishes that really let you taste the greens: Curried Greens with Golden Onions and Cashews - it calls for spinach, mustard, and dandelion greens, but turnip and radish will work great too.
Turnip Greens and Potato Veloute
- it calls for turnip greens, and mentions in the notes that radish greens work well too.
cooking.stackexchange.com/users/1672/Cascabel
Radish greens are edible and can be used in salads, although they aren't to everyone's taste (my brother hates them).
Or you can try them in a stir fry or cook them similarly to spinach.
I wasn't sure about turnip greens
so I looked it up, and they're certainly edible as well!
I'll try some next time
I have some!
cooking.stackexchange.com/users/5162/nixy 
Radish greens can also be used in soup, especially blended soups.
cooking.stackexchange.com/users/4504/Allison
I don't know much about radish greens, but you can use turnip greens as you would collards or mustard greens.
cooking.stackexchange.com/users/2832/Sean Hart
steam for 10-15 minutes + vinegar and/or (vegan) butter == yum!
(i can't imagine cooking greens for 30+ minutes -- what would remain after so much heat?)
cooking.stackexchange.com/users/2301/Nathan
Unbidden, my garden produced a super-abundance of a wide variety of mustard-y and radish-y greens this summer: leaves with bold but not overwhelmingly strong flavor, hairy undersides and juicy stems.
Determined not to put them in the compost pile, I first tried but did not enjoy the classic Southern US preparation with ham hocks.
Shout-out to Vivian Howard and crew for acquainting us with the history and uses of "Southern" ingredients in "A Chef's Life."
In the follow-up series, "Somewhere South",
Rakesh and Archna Anand prepared Saag Paneer using a variety of available greens.
Saag is a "curry" of pureed cooked greens, while Paneer is home-made cheese for which you can substitute cheese curds (aka "squeaky cheese") or feta.
(Note: Palaak Paneer, perhaps more familiar to non-Indian diners, is similar but uses spinach only.)
While the Anands started their Saag with a counter-top electric pressure cooker, the preparation continued on the stovetop for many additional hours.
My summer was saved by a recipe at SpiceCravings that produces delicious Saag in a small fraction of the time using only the pressure cooker.
My freezer runneth over with cup- and pint-sized pucks of Saag.
Thawing a puck, simmering in an added vegetable or protein and serving over rice delivers a quick and satisfying meal.
Though not a vegetarian, I prefer to replace paneer with pan-fried home-grown potatoes cut into 1/2-inch cubes.
(Home-grown = flavor; store-bought = bland)
For reasons I can't explain, simmering the pan-fried potatoes in the "gravy" for 10 to 15 minutes produces a whole which is far greater than the sum of its parts.
The flexibility of Saag with respect to greens, aromatics, spices and complementary simmering additions reveals it to be a kitchen MVP!
cooking.stackexchange.com/users/73476/Brian K1LI
Got a lovely bunch of radishes and turnips from the farmer's market, giant mass of greens included.    Can I use the greens for anything aside from compost?  Are they edible?  Worthy of salads? Stir-fry?
Uses for radish and turnip greens?
cooking.stackexchange.com
<vegetables><culinary-uses><radish>
cooking.stackexchange.com/questions/13297
cooking.stackexchange.com/users/438/Daniel Bingham
Radish greens are like mild mustard greens, and are good for salads, soups, stir fries and other dishes. Both turnips and radishes are edible and can be cooked or used raw in salads.
Turnip greens are a lot like mustard greens. They can be used from soups, stir fries or salads. Radish greens are edible and can be used in salads.