Oddysey does a great job here of explaining how just because D&D has a lot of rules for combat it doesn't mean it's about combat.
She touches on something which you hear a lot from newies raised on WOTC D&D which is: well isn't combat without all the feats and ninjitsu dull?
Initiative. Roll to hit. Damage. Roll to hit. Damage. Initiative.
No it is not. But, and this caveat goes for all Old School D&D: you have to have a good DM. Oh well.
Let us say you would like to be one of those good DMs? How do you make simple combat interesting?
Get yourself a simple combat-moves system.
Want to trick the catpegasus into charging the wall? I use d10 + stat vs. d10 + target stat. High roll wins. Other people might just use chances on a d6. If you trust your DM to come up with fair chances, these systems are nice and easy and fast. If you do not, do not play role playing games and make gingerbread cookies instead and then mail them to me so that I can eat them while I GM well.
Monsters love grabbing people.
Monsters are the grabbiest!
Roll to hit. Target number is as if for a touch attack or dex-as-(ascending)-ac. Success means target is grabbed. Less damage is done than usual but the target is still grabbed and can be hurt at will next round unless they do something about it.
Or their friends do--which isn't that hard, since their friends should get a +1 or +2 for trying to hit the monster while it is busy trying to rip their paladin in half. Unless it has like 20 arms...
Monsters Love Throwing People
Especially at other people.
Monsters Are Smart
Do your PCs use hit-and-run tactics, heal a lot and shoot from cover? Well so do the goblins.
They also hate your party's cleric above all other foes.
Monsters Are Dumb
Monsters (like players) often don't just sit back, look at their ability scores and go "Hmmmm...what't the most effective attack I could make?" they go "Gah! Shiny man hit Gorg! Gorg mad!"
Just saying, in a combat, "Well since Caroline keeps whaling on him, he's definitely going after her" both adds a little texture to the encounter and reminds the PCs they are acting in a cause-and-effect world.
Try To Use Your Half Of The Combat To Change The Tactical Situation Every Round
There are lots of things you can do that are mechanically identical to roll-to-hit-AC, roll-damage that nevertheless move people around and mix things up. If the bear hits the thief, see if the thief tumbles back and knocks a torch off the wall. Then roll to see if leather armor is flammable. Then roll to see if bears like the smell of cooking thief more than they fear fire.
Make The Tactical Set Up Interesting To Begin With
Pool of water, fireplace, ledge, multiple exits...if you are making a dungeon, put the monsters next to something they or the PCs can use to their advantage.
Use The Die Rolls
Take fumbles and crits and use them to complicate the situation:
If someone rolls a 20 you are totally allowed to go "Ok, hold it, lemme think..." and take a second to imagine that if the vampire was about to bite the ranger and the wizard rolled a 20 to throw the holy water than that probably means the vial of holy water went right into the vampire's mouth and so the vampire has a throat full of holy-water-covered-broken glass going on.
If someone rolls a 1 while charging the lizard man this means s/he's charged right past one, tripped over a lizard tail and landed in the pile of lizard eggs. Oh they are terribly upset now--but they don't want to break the eggs either...
Monsters Make Noises
A strangled gargling sound can go a long way, I find.
If you ever feel like breaking your players' hearts--have Orc A suddenly say Orc B's name as Orc B is dying. The effort-to-effect ratio is staggering.
Fear of Death Is The Mother Of Invention
This is standard OSR cant but it's true: a scared player is a thinking player.
If You Have To Say "No", Explain Why
Sometimes a player has an awesome plan and it should work. Sometimes a player has an awesome plan that is not going to work and why this is so would be clear to their PC.
Explain why it won't work in enough mechanical detail that the parameters of the problem are clear. "We need something stronger than silk, hmmmm...."
"Ok, you're grappling the plover, you have one leg in its mouth and you are in the lava" Players are easily confused in the hurly burly of the speedy exchange of blows.
Everybody knows a good DM narrates combat, but remember sometimes you just have to say it all again so that everybody is always in it.
Is It Easier Than Backstabbing?
This is standard: A thief gets +4 to backstab an opponent who doesn't know s/he's there. Basically any bonus that you give players or foes in combat should use this as a baseline: is this easier or harder than backstabbing some goggle-eyed cluelacker?
If You Can Remember It, You Can Have It!
You want to keep things moving.
"Is it d6 or d8, can't remember...."
"Well I'm telling you it's d6 since you didn't write it down"
Next time they'll write it down. In big red letters.
Relax, You Can't Do It All
(DM Energy + DM Attention) Divided By (# Of PCs + # Of foes) is a limiting factor here. Sometimes you just have to go "12? You miss. Next!" and move on to something you can do a little more with.
P.S. Answer to yesterday's Latin American literary mystery is coming, be patient.