How to get to New Vegas quickly

Basically, go from Goodsprings to Sloan to Neil's Shack. Then very carefully sneak around the outskirts of Black Mountain. In order to do this, head north from the Shack, keeping the winding path on your right. Don't go down off the ledge until you run into a mountain wall. Then carefully drop down onto the winding path, but at the switchback (where the path curves back towards the south) go off-road, past the NCR corpse and the bear traps. Dodge the falling rocks. From there it's a straight shot to Repconn Headquarters, and you're in the clear.

Then comes the part where you actually have to get into New Vegas. You have numerous options.

1. Have 2000 caps. The most straightforward way. Go all the way around Vegas and enter from the north, through Freeside. Present 2000 caps to the Securitron guarding the gate to the Strip. Note that the Securitron does not actually take your caps; it just makes sure that you have 2000 caps.
2. Have 80 Science. Talk to the Securitron guard and override it. You're in.
3. Kill all the Securitrons. Not recommended.
4. Obtain a pass by going to the King's School in Freeside (2nd segment) and doing the Kings' first quest, G.I. Blues. (Note that this is likely to negatively affect your NCR reputation.) You can pick from several different rewards. Choose access to the Strip.
5. Obtain a pass by going to Mick & Ralph's in Freeside (1st segment, near the East Gate). Pass a 50 Speech check to make the pass available. From there you can pass a 50 Barter check to get a discount, or have a high NCR reputation to get the pass for free (you still need the 50 Speech, though).
6. Sneak into the Strip via Camp McCarran. Instead of heading north from the Repconn building and passing around the east side of Vegas, head west until Vegas is directly north of you, then head north. You'll run into Camp McCarran. In order to access the monorail, you'll need one of the following: 
1. a high NCR reputation (there are quests you can do there) 
2. an NCR uniform
3. the patience to wait until the shift at the monorail terminal changes.