I'm about to start up a brand new game and I want this character to start out with his Unarmed skill tagged and the majority of my early skill points going into it. Where can I find a decent Unarmed weapon in the vicinity of Goodsprings or surrounding locations?

Closest unarmed weapon that's guaranteed to be there (ie set in the world and not randomly generated vendor/container loot) are the Brass Knuckles laying on the floor by the Prospector in the jail cell at the Nevada Highway Patrol Station south ont he road fom Primm. You've got a decent chance of finding hand wraps or boxing gloves in the locked locker at Jean's Sky Diving (key is on the table by the radio) south east on the road out of Primm but no guarantees.

I think...( and yes, it's painful ) a Bladed Gauntlet is always located in the Mojave Express, in Primm. IIRC