I have four cat, a mom and three of her nearly three year old kittens. The mom HATES one of her sons, yowling and taking swipes at him whenever they're in the same room, and barely tolerates the other two kids. One of my boys gets in moods where he'll chase his sister around the house while she's hissing and yelling at him, usually until she crawls under the couch.
Cats are like people, with no impulse control. They each have their own personality, and personalities do not always mesh. Cats are also territorial and have a strong sense of hierarchy. Put that all together, and you can have a pretty volitile mixture.
Right now, your new cat is trying to fight for a higher place on the totem pole, and the others are fighting to keep their places. They're also trying to work out who can/will be friends with who, and who gets to claim which favorite spots in the house, and even the humans' attention. It'll take awhile for everything to settle down, and there will probably always be random spats. They may never be friends, but they'll probably learn to mostly ignore each other.
As long as there's no blood or real injury, just let them work it out. Fur will fly, literally, during scuffles. Teeth and claws will rip out fur, and a lot of cats shed overmuch when stressed, so don't panic if you find tufts of fur lying around.
If you're worried about injuries, you can trim or cap their claws. Wrap the cat in a towel, to hold them still and protect yourself, and pull one paw out at a time to trim the claws. Make sure to only trim the very tip. The darker vein further back, toward the thicker part of the nail, is a nerve, you DO NOT want to hit that. You can also buy little plastic caps that glue onto claws. Either method will probably have to be redone every week or two.
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