Grass-fed beef is leaner than grain-fed, which means it takes a little finesse to cook it right. While medium or medium-well burger patties might be to your liking generally, grass-fed patties cooked to those higher temps will come out dry. And it's more expensive, so you want to make sure you treat it optimally.
As far as doneness goes, most grass-fed cuts meant for grilling or sauteing do best when done medium-rare. Tougher cuts with more connective tissue like chuck or shank will benefit from the same low-and-slow moist heat braise and stew techniques you use for grain-fed beef.
But since summer's about her, the time is right for steak. Check out this recipe for pan-seared grass-fed steak, which emplys a time-honored restaurant technique of basting the beef in the pan with butter and aromatics as it finishes to add flavor and ensure moistness. If you're grilling the steaks, you can use the same method, just place the almost-fully-cooked steaks in a warmed cast-iron pan on a grill set up for indirect heat on the cooler side, add your butter and additional ingredients, and baste for a minute or so until done.