Well, I think the first thing to realize is that weapons were specifically developed for different fighting styles, so to begin with you’re going to want to have a good idea of which weapon/weapon(s) it is your character uses and the fighting style that they will be fighting in. Here’s a few links I found that may help:
Spears: (FYI, actually the best common weapon in history.)
Sword Fighting Techniques (Might be a decent starting point for further research)
(You’ll have to do more research on your own since this is a huge subject and different swords can have very different fighting techniques. Be sure you read about the sword you’re interested in using thoroughly to be sure you have your character using it the right way.)
Bow & Arrow/Crossbow
One thing to note is that the bow and arrow has never (or not often) been used in direct combat. The reason is pretty simple. If you can take down your enemy from a safe, secure position far away from the main fight, why wouldn’t you take that for granted? You get a little more flexibility with a crossbow. One thing that always amuses me about crossbows, by the way, is how little they’re used in fiction. They’re just as deadly as guns with the added bonus of being pretty much totally silent.
There are a million different types of guns, and each of them has its own use. One thing I’ll note here is that guns require a ton of research, especially if your story is set in the “real” world, so be prepared to take on that task if you have them in your story.
In a lot of popular fiction, you’ll see weapons like whips being used in combat on a regular basis. Whenever you feel the need to bring a more unusual weapon into your story, always stop and think about whether or not it’s actually practical. Nothing breaks me out of a scene quicker than thinking, “Yeah right! Like that would ever happen!”
Also pay attention to whether two weapons would be practical to use against one another. A smart fighter isn’t going to stand around and try to hold his own when he knows he’s seriously outclassed on the basis of his opponent’s weapon alone.
One more thing to note - a character is never going to be able to pick up a weapon they haven’t been trained to use and have the ability to fight with it. They may be able to at least hold their own if it’s similar to a weapon they do know how to use, but they certainly aren’t going to be an expert with it. Using any weapon requires extensive training, otherwise the weapon is going to be a hindrance more than it’s going to be an asset.
As for actually writing it, remember: you do not have to offer your readers a blow by blow account of what’s happening. When you write a fight scene, it should essentially be a highlight reel of the biggest and most badass moments that happen during that fight.
Write in shorter, faster-paced sentences to build tension, and pay attention to what your POV character would actually be able to notice.
For example, they may feel themselves get cut, but they’re probably not going to have time to take notes of the severity of the injury or the specifics of it. They’re probably not going to notice their best friend getting cut down on the other side of the room, although they might hear an ally scream for help. In a serious fight, your character is going to have to compartmentalize and focus on the most important thing - survival.
I may come back later today and try to comment a bit more on writing action scenes, but I think these basics are a good starting point. Above all else, just be realistic about what your characters can do and what they can pull off in a fight.