Lower back exercises: deadlifts, good mornings, back extensions/hyperextensions. Squats aren't really a lower back exercise, but they pretty much work EVERYTHING in your body. You have to make sure that you're doing these lifts with proper form however. It's crucial. Deadlifts and squats are great exercises, but doing them wrong can mess you up for a long time.
On th P90X note, I did the program a year or two ago and it is really good. However, I really have to mention that people may not be painting the clearest picture here as to how hard it is and how much dedication it takes. This program is going to take about 1.5 hours of your time daily (6-7 days a week), so just be ready for it. The workouts are also very intense (you can do them at a slower pace if you want). Overall it's a great routine for getting back into shape for sports and the like. I'm not sure whether I would recommend it long-term though, since I find it to be a bit of an overkill. The YogaX routine they have (as well as the stretching routine) are really good. They aren't very many peoples' favorites (I was almost never motivated to do YogaX, but forced myself to anyway), but they really do you a world of good. As far as eating healthy and sleeping goes, that really has nothing to do with P90X. Any exercise program should be supplemented with those two things, and they should actually be an every day thing for anyone, gym or not. Eating healthy and sleeping right are just things that you should do if you want to be healthy.
For golf, I'd agree with the more explosive exercises as well as focusing on your core and rotator cuffs. Core exercises would be: squats, standing press, bench press, deadlifts, bent over rows. You can also throw in power cleans which are an amazing exercise, but you really have to learn the technique (which isn't easy) properly. Build a routine around those core exercises with some secondary stuff like bicep curls, dips (awesome exercise), pullups/chinups (awesome as well), lateral raises, leg presses/curls, calf raises, and some ab exercises.
Typically you have two choices. Either do a full body routine (which usually take longer, but are a great choice for beginners and people that know what they're doing), or split up your routines into muscle groups (i.e., one day for Back and Biceps; one for Chest, Shoulders and Triceps; one for Legs).
I personally don't do squats anymore (yes I know it's THE exercise) since my legs grew so much that I had to get a whole new wardrobe and couldn't find any pants that would fit or look good. I stick to cardio and sports mostly for my legs now.
In sum, I think what will help your golf most are core exercises, as well as a routine which promotes flexibility (such as Yoga or stretching).
Cheers,