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The sheriff in Redding can give you a badge as a deputy and four quests. When you've completed his first, see him again for quest #4 below, then #2, then finally #1. This is the order the game lists them, so I'm in general trying to be consistent with the game.
The mayor can sell you a deed to the Great Wanamingo Mine for $1000, and will pay you back 150% if you clear the mine of the aliens. The well to the south side of Redding can take you into the mine. Beware of the wanamingos, as they have a high damage resistance and hit multiple times. Come prepared with a burst weapon and save often through your journey here. I found it easier to come back a little later when I was more prepared--they aren't the most difficult enemies in the game--and complete the quest, but it is doable when you first get here. When they're all defeated, including the eggs near the elevator, you'll get an xp reward, and can then return to Ascorti the mayor for your return of investment and more xp.
There's a caravan on the northern side of town, and you'll find Painless Doc Johnson to be in on the dope about the Jet problem in Redding. He can also perform combat implant surgery if you've learned about it from Vault City, for fairly cheap.
Quests & Opportunities
Here is a list of some of the quests and other opportunities in Redding, each with a brief synopsis containing at least one way to solve it. The remaining ways, and possibly other minor quests I will leave up to you.
1. Kill Frog Morton:
The sheriff's last quest for you, you'll find his gang on the western side of town. After defeating some rats, begin your attack on the gang... which can be fairly challenging if you're still around level 11 or so, and be sure to loot their bodies for nice weapons and ammo for this stage of the game. After returning to the sheriff for your formal reward, he'll mention the morton brothers, and you may have encounters with each of the three throughout the game as you travel, but they're nothing to be worried about.
2. Find out who cut the whore:
This is the sheriff's third quest. Talk to Lou at the Malamute saloon for a tip. Now head to the Morningstar Mine and confront Obidiah, at which point he'll confess. Take him into custody for the best reward.
3. Find the excavator chip:
Dan McGrew of the Morningstar Mine or Marge LeBarge of the Koko Mine are interested in these, but you'll have to make it down to level 3 of the Wanamingo mine... and deal with the tough ones (see top for a brief description). There are two chips here, one on the ground and another on the excavator. Pick either one up and return it to either person for a reward.
4. Break up the bar brawl:
This is sheriff Marion's second quest, and it's up to you to mediate between the two altercationists in the bar. You'll have to threaten violence to solve this one, as simply shutting them up won't help. When you've taken one or the other, or both to jail, return to the sheriff for a reward and your next quest, #2.