Shield and health
Your shield is rechargeable of course, but preferably don't let it get so low that it starts bleeping, else you'll be in imminent danger of losing health. To minimize damage when having a firefight out in the open, move around as you fire; don't stand still. The enemy are poor at leading their target, and will often miss if you move across their line of fire, perhaps dodging left and right or circling around. Of course, good use of cover is another basic skill you want to develop. The less of your shiny tin hide you expose, the better.
A more realistic style of play
It would be pretty easy to lapse into a habit of thinking "I can afford to let my health take a bit more of a knock because there's a medkit coming up shortly". To my mind this is rather false, resting on advance knowledge that you really wouldn't have. It also seems a bit lazy. If you want to adopt a more realistic style of play, or if you want some added challenge, you need to make a conscious effort to reduce any medkit dependence. This basically means having the discipline to avoid any risky gung-ho stuff, and instead being patient and careful. The ideal would be to get through battles with no health damage at all. That's quite a challenge on Legendary, and it may inspire you to refine your technique or develop new combat methods; see the Perfection challenge.
Increased caution on low health
From experience, I get the distinct impression that the lower your health is, the faster the enemy is able to cut through your shield. If this is really true, it's another reason to try hard never to let your health fall, because it means you're going to be able to take on less, despite having your shield up. In any event, when your health is down to only a few bars, you're living on something of a knife-edge because those bars could easily get wiped out with a burst of fire that breaks through your shield. So when on low health, you should be extra cautious and not be lulled into a false sense of security by your shield.
Enemy overdrive when you're in trouble
When your shield is down, or possibly even when it's merely getting close to failure, enemies seem to get significantly more aggressive. It's as if they've sensed you're in trouble, and are mercilessly trying to finish you off while they've got the chance. It's a commendable bit of design from Bungie, making enemies seem all the more intelligent. This 'overdrive' behaviour, which I've found particularly apparent under the pressure of enemy swarms in the megabattles, can quickly make your situation desperate. In particular, Grunts and Jackals can become noticeably braver than usual, often pressing on with an attack even after the death of a nearby Elite, something that would normally make them scatter in panic. Needless to say, this is all the more reason to try to keep your shield intact, and to buy time to recharge when it's in poor shape.
When your shield is damaged, it seems to take about 7.2 seconds until it can start recharging (on a PAL Xbox at least). The subsequent recharging rate is dependent on difficulty level however, becoming faster as the difficulty level rises. For a fully depleted shield the recharging time is about 9.3 seconds on Easy, 4.6 on Normal, 3.1 on Heroic and 2.3 on Legendary. I have to say, whenever I play on Normal rather than my usual Heroic or Legendary, I really notice the slower recharging and find it quite frustrating in hectic megabattle situations where the pressure keeps piling on. I'm not sure what the rationale was for making it slower for lower difficulty levels. It doesn't make sense to me, because slower recharging makes your life harder of course.