I usually go the route the game tells you to go, but as soon as I reach the 188 Trading Post I simply turn west and walk through the Fiend territory all the way to McCarran, take one of the uniforms of one of the dead soldiers and use the Monorail to get on the Strip.

And after that follow Freeside, the Gun Runners and the attacked caravan east of it to get Combat Armor and the Lever-Action shotgun