As others have mentioned, being healthy is a somewhat subjective thing to aim for. So instead, I'm gonna aim to answer, as much as I can, your definition of health:
This is a particularly subjective one, and all I can offer is impressions from what I've read.
There has been a fair amount of research linking inflammation to disease, and several authors have already written a fair amount about regular exercise to reduce chronic inflammation.
There doesn't seem to be one particular exercise regime resulting in reduced chronic inflammation that I could find, it was more a case of "do exercise stuff" where exercise stuff is taken to mean resistance training as well as cardio type stuff (both aerobic and anaerobic).
As Dan John is a big fan of saying, the keys to living to a ripe old age are don't smoke, wear a seat belt and learn to fall (amongst others). The first two seem pretty self explanatory, but the last one sometimes confuses people. Basically, the older you get, the more likely you are to have a fall, and if you land badly, that's when things go snap.
Ok, so being slightly more objective, heavy resistance training increases bone density, and the more dense your bones are, the less likely they are to snap upon impact. Along the same lines, if you carry a fair amount of muscle on your body, you effective have a form of natural "armour" between your bones / insides and whatever it is that you decide to impact on.
Again, this is rather subjective, mainly because it depends what hurts!
Speaking from personal experience, I've found that getting myself stronger and constantly working on my mobility has helped keep me reasonably pain free for the past several years.
I've fixed back issues by working on my deadlift strength and hip mobility; I've fixed knee issues by working on my squat technique and hip mobility; I've fixed shoulder issue by building strength around the shoulder joint (Turkish get-ups) and loosening up tight pectorals.
If you want general suggestions, go with what Dan John suggests and strengthen the phasic muscles, and stretch the tonic ones.
Mentally and physically better quality of live
I'd suggest reading this first.
There's a certain joy in being strong that isn't easy to express, because it's not until you realise what it's like to not have it that you realise what a great thing it is.
It's those times when you open tough jars without thinking (sometimes by them being handed to you by someone else), that time you helped a little old lady load something into the boot of her car, the time you helped your friend move house, listening to the laughter and joy as you lift your grand kids up onto your shoulders, that time you had the strength to catch your pregnant partner when they passed out, that time you helped your dying father sit up in his hospital bed so he could watch the sun set one last time.
Had I not have had the strength to do any of those at the time, it's not just my quality of life that would've suffered.
A shot at surviving the zombie apocalypse
Despite rule #1, I'd say that being strong is one of the key factors here. If you can't break down doors when needed, fight off the undead craving your flesh, or pull yourself up onto that roof to get away from the hordes, then your chances of survival are pretty slim.
I've always valued strength above all else. There is something to say for putting on muscle and looking good naked, especially when you're younger, and it can honestly do wonders for your self confidence if you don't feel the urge to kill the lights before stripping off in front of someone. But (NSFW!!!) there's something beautiful about watching how someone strong enough to have full control over their body moves (NSFW!!!).
Do sets of 5 for the big lifts (deadlift, squat, overhead press, chin up, row) and sets of 10 for the not so big lifts (curls, lunges, triceps extensions).
Finish off with a few sets of heavy carries for distance, and spend a little time stretching where you're tight. That should see you pretty well.