Know Your Role!
Every ship has a role. Know what you are flying, what exactly you are supposed to do on the battlefield, and exactly how you are supposed to survive. Fit your ship accordingly. My rules may not be the exact role of your specific ship, so use my guide, but use your head as well. The best way to learn your exact role is by looking at the ship bonuses you get from your skills under the description tab on your ship.
Frigates
Frigates are like the grunts in a fleet. Generally very disposable, and can get in and out of combat quickly. They don't get the glory, but without them the rest of the fleet would be pretty useless. Frigates are fast and can orbit enemy ships at extreme speeds. Frigates are quite small. To a Battleship a frigate will look like little more than a slow orbiting drone. Because of their great speed and small size they also have small guns, and defenses. Don't try to go toe to toe with anything other than another Frigate or Destroyer. The Frigate's main roles are killing other frigates, and Tackling. Tackling is orbiting at close range and getting electronic warfare down on ships. Generally that consists of a warp jam and sometimes one other electronic warfare device. Always fit an afterburner or micro-warp drive on your ship. Speed is always your number one priority. Ideally your frigate will be orbiting at 4,000 m/s at a range of 6km. That way you are out of the range of smart bombs, and moving faster than most missiles. Don't be afraid to run if necessary. You are small, fast, and take the longest to target of all the ships. That is part of the job. Drones are going to be your biggest threat. If any ship begins sending drones on you GET OUT. Even 5 small drones will be more than enough to take you down. Just remember. You are small, fragile, and there for support. Run when necessary and support your bigger ships.
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Destroyers
Personally I am not a fan. They are small, slow, and blow up almost as easily as Frigates. They only have one role-blow up small frigates, other destroyers, and drones. Make sure nobody fit in a frigate is putting any electronic warfare down on your allies. You need those big ships to take out their big ships.
Cruisers
Cruisers are solid medium sized ships. They are more expensive than a frigate by a long shot, but still generally viewed as quite disposable. They can hold their own in a fight much better than frigates, but still rely partially on speed. They can be fast enough to keep up electronic warfare, and big enough to harm even the biggest of enemies. These aren't the heaviest hitters, or the fastest movers, but they are very versatile. Their medium guns and drones can be plenty to kill a frigate, and enough to still rough up a Battleship with some help. Don't forget to orbit the bigger ships.

Battle Cruisers.
Bigger than Cruisers, but still in the medium category, Cruisers are where the guns start to get good. They are effective against almost any ship. Their drones can tear frigates apart, and their huge amount of guns are effective against almost any ship. Though faster than battleships, they are generally too slow to orbit, so try to keep your enemies at your optimal range, but thats going to be about the extent of your ability. This means your ability to take head on damage is more important than ever. Electronic warfare begins to become less of an option unless fighting smaller ships that are keeping in range of you.
Battleships
The biggest non capital combat ship, Battleships are the big guns of the Eve Universe. In a fleet they can hit targets at huge ranges, and rely on frigates to keep their enemies from warping. They take a long time to lock onto smaller ships, but once they do their drones are enough to kill frigates (assuming they haven't warped out yet.) Orbiting is out of the question. Just try to keep at optimal range and unload on your targets. They will drop quickly.