I must have fought this guy 15 times when I first got the item from headmaster Cid. I still haven't beaten him, help!
As I recall, Diablos is pretty challenging when you first get him if you haven't been doing any grinding or excessive magic-drawing early on.
The key to the fight is the spells that you can draw from him. He always carries Demi, and will have either Cure, Cura or Curaga depending on his (i.e. your) level. Draw these and use them to your advantage.
The Demi spell instantly removes 25% of the target's current HP. This means you want to draw it and cast it as your first couple moves. The more HP he has, the harder it hits. In this way, you can drop him to half health in two shots.
From there, beat him down with standard attacks and magic. As a flying monster, he is weak to Wind. Make liberal use of the Cure-family spells you can draw from him.
His Gravija spell hits extremely hard, but it works the same as Demi, it just takes a much higher percentage of your health. If he puts you into position for a Limit Break, you can also do a lot of damage that way. (Remember, you can hit the Circle button to retry for a Limit Break if you're at low HP).
Finally, don't be afraid to use any healing items you have. Items are plentiful in FFVIII, and you'll be able to easily resupply later on.
The key for me was having blind (drawn from the Granaldo Boss not long before you are first able to battle Diablo). If you draw a full stock of it for at least one character, and have it equipped to ST-ATK, between attacking with that character and casting it you should be able to affect him fairly quickly (note that he has a high resistance to blind, but is not fully immune).
Once you successfully hit him with it, it is almost impossible for him to kill you - he will cast Demi and Gravija most of the time, which can only reduce your HP to 1, not kill you. Only his punch attack can kill you, and he will almost never hit with it once blinded. You can leave your characters on low health (using their limit breaks with abandon) and just Phoenix Down them if they do happen to be hit by a lucky punch. As per the other answers here, still start by spamming Demi, as that will do far more than even your limit breaks (if you are attempting this fight early-game) until his health is reduced to a more manageable level.
I seem to recall him having a not-so-fun melee attack as well. Blinding him was always the first thing I did, though it takes a few attempts to do so.
One of the tactics used by veteran players is to keep Squall and Zell at low health so you can use their limit breaks (I think that's what it's called in FF8 but it's been a while) every time they get an action.
Start the battle by casting some Demi spells as sjohnston suggests, because that is the fastest way to get his HP down at first, but then try to keep these two characters on low health with the other higher, as a dedicated healer for when those two inevitably faint in battle.
If you draw demi from him, then cast it on him from your own stock rather than his, he will immediately cast curaga on the caster. Very useful.
I defeated on first try when I received the lamp.
People here stating that blinds help etc.. but I dindt do any of that. Instead, I just made sure one ally had low HP (Graviga was actually quite helpful to reach that point) and spammed Limit Break. The other two were focusing on keep the other two of the party healthy. It requires a little luck because I was lucky enough that he only targeted the Limit break spammer with Demi, and he attacked the healthy ones.
I was surprised how easy this was. Hope you find your own way too :) No need to grind for it atleast, there are ways to beat it on low lvl relativly easy.
What I did was blind him early game, and then use the limit break slot from Selphie to keep changing the cast until the end comes out, and use that skill.