Avoid Ion Shield Damage
If your enemy is attacking with an ion weapon, consider letting your shields down. If it hits the ship while your shields are up they will be reduced in power for a few seconds. If, however, it goes through your shield it may hit another, less crucial system (or even an empty room) and your shields will be undamaged.
Power Level Your Crew
Your crew members gain valuable skills while they are in combat. Use this fact to your advantage and prolong combat when your enemy is unable to damage your hull.
A few hints:
If you have a low-level weapon that will not damage the enemy's hull keep firing (so long as you are not in danger). Your crew gain's experience with every shot. Disable the enemy's engines to maximize your combat time. This way they will not be able to escape.
If your enemy is unable to penetrate your shield just sit tight! While they futilely attack you your engine, pilot and shield crew gain experience.
If you have a cloaking device be sure and enable it when the enemy fires - even if they can't hurt you. You gain skill from dodging.
Cycle your shields during combat. So long as your timing is good you won't take damage but the crew member manning shields will receive a small boost.
If you encounter combat in an asteroid field after destroying the enemy ship, don't jump! Remove your pilot to drop your evasion to 0%. Every asteroid strike that hits your shield will give your crew manning the shield 1 experience point. On average you can max your shield crewman in 10 minutes.