Chamorro cuisine is the foundation of eating in Guam, and it's worth seeking out. The defining dishes are built around rice (red rice, tinted with achote seeds, appears at almost every meal), barbecued meats, and citrus-cured preparations. Kelaguen is the most distinctively Chamorro: meat — usually chicken, beef, or occasionally venison — marinated in lemon juice, fresh coconut, and hot peppers. It's closer to a ceviche than a cooked dish and has a bright, sharp flavor. Hineksa' agaga' (red rice) and kadu (a simple meat broth) are everyday staples. Fina'denne', a sauce of soy, vinegar, and hot peppers, appears on tables the way hot sauce does in other cuisines.
The island's food culture has absorbed a lot of outside influence — Filipino, Japanese, Korean, and American dishes are all part of everyday eating, and the distinctions often blur. You'll find spam used as a legitimate ingredient (a Pacific-wide habit), noodles from Filipino tradition showing up alongside barbecued ribs, and Japanese ramen shops next to taco trucks.
The Chamorro Village Night Market in Hagatna on Wednesday evenings is the best single place to eat local food without sitting down at a restaurant. Try the barbecue, the red rice, and whatever's coming off the grill. For sit-down Chamorro cooking, Meskla Chamoru Fusion Bistro in Hagatna is well regarded for its modern take on traditional dishes. Da Local Grindhouse in Dededo does excellent beef kelaguen and no-nonsense Chamorro barbecue.
The Tumon Bay tourist strip has the full range of international restaurants — Japanese, Korean, Italian, American chains — but most of it is generic. Eat there if you need to; eat elsewhere if you can.
Coconut and tropical fruits show up throughout — coconut milk in stews, fresh papaya, mango in season. Local fruit stands along the southern villages are worth stopping at.