Series Weapons

Look here or

Apparently old guns die hard - or classic designs never fade.
In the 26th century, firearms seem to look exactly like the ones we have now. Mostly. One thing that has changed is the use of caseless ammunition on many of the 'mid-range' weapons -- there are still standard 'cartridge' weapons with the expected number of shots, but many weapons seem to use caseless ammo, which means no brass flying out of the side of the gun, a chemical propulsion system -- which is why it sounds funny when many of the guns are cocked or fired, and increased ammo capacity and rate of fire on guns that you wouldn't expect it from (Wash, for example, fires what looks like a revolver on what can only be described as "burst" mode during the scene on Serenity in Heart Of Gold).
Jayne's favorite pistol (he's even shown cleaning it at the table) looks to be a Le Mat revolver (from 1860) with a barrel shroud. He carries it all the time, but is only shown shooting it in the "Out of Gas" flashback. Lawrence Dobson used a 1921 Astra 400, or Astra 600 (depending on caliber) in "Serenity." Mal's revolver (which has been confirmed as molded aroung a five-shot .38 caliber Taurus model 85 revolver) is a (likely modified) officers' sidearm from the Unification War, almost certainly the same one he packed in the Battle of Serenity Valley. Notice whenever Mal cocks the hammer of his gun, there's the sound, but the hammer never comes back. Even in close-up, you can see the thumb piece is stationary. Inara's pistol is a Ruger. In the 1950's and later, a factory compensator was available, on the Mark I, for target shooters. Zoe's Mare's Leg might be a '92 Winchester, or a '94 (the '92 was in pistol cartriges (.25/20 to .44/40) and the '94 is in rifle cartridges and pistol cartridges. Pistol cartridges are the only way to get 6 rounds in that button magazine). The sheriff in "The Train Job" carries an old Remington Rolling Block rifle. They've been making those since 1867.

Alliance Assault Rifle
Anglo Sino Alliance troops are seen with a variety of rifles, either bullpup or the more conventional rifle layout.
Several props have been seen to be used. The bullpup example being the SA-80(approve sites) bullpup assault rifle, the standard-issue assault rifle of the British Army(approve sites) and the rest of the British armed forces. The more standard rifle being a Heckler & Koch G36(approve sites) or its carbine variant the G36K.
Best examples can be seen in the Serenity Pilot and The Train Job.