You fought the law, and the law won! Your missions come in sets of 3, and there is no break at all between them.

Highwayman
Right after the bridge battle in Dodge City, you can talk to Pat. He says to keep the peace while he's occupied. First, kill the highwaymen - they're over to the right. There are 3 guys on horses. You can kill them with your rifle pretty easily. Now you're told there's a poker brawl in the alley, with no break after the first part ended. Slay them. You go right into an opium peddler, and you're told to talk to the bartender. Note that he's not at the Alhambra, he's at another spot. You're told that the pusher has bodyguards. Go slay them all.
Reward: gunhand +4, quickdraw +5, melee +2, cash $10

Whore
Immediately after the first mission above you can return to Pat Denton. He'll now make you a deputy. Your first task is to save a whore. She has a health bar. Kill all the bad guys, it's pretty straightforward. Next you get a hostage up in a tower. You can shoot through a hole in the fence. The final guy has a girl hostage - shoot him out from behind her. OK, now there's a hold up going on across the street. behind the saloon. The victim has a health bar. He has almost no health left, but if you shoot quickly, he's ok.
Reward: gunhand +4, quickdraw +5, melee +2, cash $10

Campfire Gang
Third in a row for Pat. He says to keep an eye on things. First, eliminate the campfire gang. They're hanging out at the far right of town. There are only 4-5 bad guys. When you're done with that, there's a powder keg delivery down by the ferry. Run down to the dock and grab it. Now you have to head back to town - but of course you have enemies ahead. When you see the red dots, put the keg down and shoot the bad guys. When it's clear, then go back for the keg. Run to the X again. Once you get to the middle of town, you meet another wave of bad guys. Shoot them out. Drop the keg in the ring. Next, raiders are riding in from the east. They're fixing to shoot up the town. Get out your rifle; they ride straight at you and it's not too hard to pick them off.
Reward: gunhand +4, quickdraw +5, melee +2, cash $10

Miller Gang
This is your first Empire mission. Go to the ambush area that is marked; go around to the X and up on roof to the TNT area. Throw the barrels down as instructed. I believe you have been training with Yoda - the barrels slide towards you strangely. When the're all down, shoot down at them. Defeat the miller gang - complete.
Reward: gunhand +3, quickdraw +5, melee +5, cash $10

You get these Federal Marshall missions after the first Hoodoo mission

Smallpox Vaccine
They need the smallpox vaccine back. Thieves are on rooftops. You can headshot from the ground with your rifle. If you need to go up on the roof for a final one or two you can't get an angle on, hop up on the mini-wall by the lower building and then up. They really don't shoot back at you so it's pretty easy. The vaccine is left behind up on a high spot - walk across the boards on the roof and up the ladder, 2nd ladder and up. There's a gatling gun there too :)
Reward: gunhand +3, quickdraw +5, melee +5, cash $10

Chinese Hostages
Apparently a Chinese fellow and his buddies are held hostage in the badlands. Get a horse and ride on out. When you get there, sneak in, be careful and head-shot them while avoiding the TNT. sneak down the mine shaft and peer around the corner and shoot the guards. Again, be very careful with your shots. The last guys are up a side alley hiding. There are 2 guys there each with a hostage. When you clear them, complete.
Reward: gunhand +3, quickdraw +5, melee +5, cash $10

Missing Coach
Go find the missing coach. First, get a horse. Follow the X down to the driver area. Leave the horse a little ways away so it's out of the way. Now start shooting as soon as the bandits turn red. They'll be all around you. When it's clear, go talk to the real driver. You now need to get a guy to the hospital. He says the TNT is above the boulders. Go up on the ledge above and throw down the TNT. When it's down, go down and get them near the rocks. Finally, go back by the coach and shoot it. complete.
Reward: gunhand +3, quickdraw +5, melee +5, cash $10

Gun Walkthrough