Plan low-key dates where your guy isn't feeling pressured to be the life of the party. He'll probably enjoy an evening in watching a movie to a high-octane night out with a large group. Do things together that will reinforce his self-esteem, like whipping up a delicious meal.
Resist the temptation to obsessively fill silences with chatter. Shy guys are often comfortable just sitting with you and enjoying your company. In fact, shy people often compare themselves unfavorably to outgoing, chatty types, so your monologue might be making him feel inadequate. In quiet moments, hold his hand or smile at him. Ask him questions about himself -- his favorite book growing up, or memories of his grandfather -- that will draw him out of his shell.
Introduce your friends and family slowly. A quiet dinner with your parents will be less intimidating for your guy than a family reunion replete with cousins and your loud, obnoxious Uncle Bob. Invite your best friend along for mini-golfing with you and your guy, rather than bringing him along to her birthday bash. Help your guy open up to your loved ones; tell your family that he's an accomplished guitar player rather than waiting for him to volunteer the information.
Remain patient and supportive during social situations. Hold his hand to reassure him at your office holiday party, and refrain from nagging him about his shyness. Avoid comments like "You didn't say anything at all to my dad," as they're destructive and will make him feel inadequate.
Reinforce him with positive comments. Tell him things like, "All my friends were raving about what a charming guy you are," so he'll feel accepted and liked. Compliment him often. Tell him that he looks handsome today, or that you really appreciate how helpful he is with your car. Compliments will counteract feelings of inadequacy and make him feel loved and appreciated.