I also loathe 'sport specific' training.
Personally I like free weights with emphasis on Olympic lifting. Focus on the basics: the deadlift, shoulder press, squat, and bench press. Body weight movements such as pullups, situps, leg lifts, pushups help too.
Flexibility is also very important. Weight lifting alone may not help improve that area. There are specific mobility drills to help improve hip mobility, shoulder mobility, ankle mobility and so on. Improving flexibility helps all areas of fitness. You might be surprised at how limited you are, not by your strength, but by tight quad muscles and hamstrings.
There's no 'one size fits all approach' to fitness. Good trainers consider the age, injuries, and realistic outcomes for their clients. Good programming with emphasis on safe and proper technique cannot be overstated.
Without a good program (and accurate notes and records of what you lifted) the progress will not come and frustration will boil to the surface. Don't half ass it. Get a notebook, get a good coach/program, write down your goals and bust your ass to break through to new barriers. A focus on general fitness will help make you look good and feel better as well as being able to tackle a variety of different skills (not just golf). So don't be scared to try new things: sprint sessions, bike rides, weight lifting, swimming, climbing, gymnastics, conditioning drills, and so on. Mix it up - it keeps things interesting.
Also realize that 70% of all fitness related goals are tied to your diet. "You can't out-train a bad diet." There's most certainly a synergy between the food (fuel) and the exercise (spark).
Erik