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You'll find several quests in NCR and can recruit the robodog NPC if you have a fairly high science skill (see quest #2).
Your first time here, you'll enter in the Bazaar area of NCR. The merchant you'll want to see--only during the day though--is Buster. Watch that you don't bother him during the night, however, as it could lead to unfortunate eventualities and he's pretty good for your energy ammo needs.
In the back of the bar at night, you'll find Merk. If you ask him about work, you can get quest #7... but it costs you a little every time you ask him something.
During the day in NCR, you'll find Dorothy in the northeast corner outside the power station. Save your game because during the conversation you can be teleported into helping talk Jack--the insane man with a bomb inside--out of killing himself. You'll need a high speech of 80% to pull it off. Jack is involved in quest #11, so it's recommended that you allow him to live. You also get an insane amount of xp for completing this task for Dorothy without violence so it's worth it.
One of the best things to do in NCR is to find the Hubologist and have a chat. She's located in a building near the Cantina and she can raise your Luck by two or reduce it by one, so save again before your conversation. She'll also give you quest #6.
On the western side of the Bazaar map, you'll find a guy who can outfit your car with a Blower for $1000, a useful thing to do.
When on the city map, it's important for questing reasons to find Dumont the Sheriff on the south end and ask about work. He'll let you know about Westin and President Tandi. You may already have heard about Westin from Vault City's quest #9 but it's likely you haven't.
Quests & Opportunities
Here is a list of some of the quests and other opportunities in NCR, 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. Retrieve papers from Dr. Henry:
Merk's second quest (see #7) is to find these papers. Henry's place is south of Duppo the merchant and the papers are in his desk. While you're there, think about picking up quest #2 if you've got a high science skill of 80% or greater.
2. Test mutagenic serum on a super-mutant. Try Broken Hills:
Speak with Dr. Henry in the building south of Duppo the merchant with a science skill of 80% or greater, and ask about Vault 13. You should then be able to discuss cyber-genetics and receive this quest. The easiest super mutant to test the Mutagenic Serum you received from Henry on is in the Bazaar area in the bar, not Broken Hills. Simply use the serum on the mutant and report back to Henry with your finding and you'll receive both xp and if you want, the robodog NPC.
3. Deliver Westin's holodisk to Lynette in Vault City:
Once you've taken quest #9 in Vault City Westin will give you this one. To meet Westin normally, talk with Sheriff Dumont inside NCR. At the completion of this quest which is straightforward, if you've been nice to Lynette from the start in Vault City and have a high Charisma or 75% speech, she'll possibly make you Captain of the Guard and you'll net more xp.
4. Get the map from the NCR Rangers, for Vortis the slaver:
You might want to do this quest before #5, as it requires that you kill the slavers. You'll find Vortis in the Bazaar area. Talk to him during the day after noon, else you could be attacked during the night. Ask Vortis for work and you'll get this quest. To complete it, head on up to the Ranger's base downtown, into the garage and grab the map off the table, and return to Vortis.
5. Free the slaves in the slave pen, for the Rangers:
Talk to Elise just north of the Brotherhood outpost about joining the Rangers and you'll get this quest. Head to the Bazaar and into the slave pens, kill the slavers and use the terminals to free the slaves. For this you'll get a fancy Ranger pin and decent xp.
6. Deliver Hubologist's field report to AHS-9 in San Francisco:
You'll need a fairly decent Speech skill and Intelligence--use drugs if you have to in order to raise it to 8 or higher--to pick up this quest from the Hubologist here in NCR, who resides near the Cantina in town. Ask her about auras and then how she is doing and she'll give you the quest. For delivering the report at the Hubologist base in SF, you'll get a smidgeon of xp and that's about it.
7. Kill the Hubologist in NCR for Merk:
When you ask Merk at the back of the bar for work, he'll give you this quest. Before you complete it make sure you've saved your game and talk to her to raise your luck and receive her quest as well, #6. The easiest way to kill without remuneration is to use Super Stimpaks on your victim and then to wait 10 minutes on your Pipboy. The hubologist will take a few of them.
8. Stop brahmin raids:
To meet Westin talk with Sheriff Dumont inside NCR. Ask Westin about Vault 13 to receive this quest, then Felix outside the door and you'll be teleported to a brahmin pasture. Wait there until a couple deathclaws show up and leave, and then head back to Westin and tell him about posting a guard for your reward.
9. Eliminate Mr. Bishop:
If you've been to New Reno and picked up quest #7 there to kill Westin, you can double-cross Bishop by accepting Westin's promise to increase the payout. The result no xp and obviously no more Bishop quests, so do this only if you're comfortable with that.
10. Retrieve Parts/Gain Access to Vault 15:
First talk with Sheriff Dumont, who can tell you about Tandi. Next, visit the Congressional building and make your way to Tandi... the very same girl you (may have) rescued in Fallout 1. She'll give you this quest. Now set off for Vault 15 which will appear on your World Map if it isn't already. Complete the quests there (and don't forget the spy holodisk if you're capable--see Vault 15 quest #1) and return to Tandi for a sizeable cash and xp reward... if you didn't first ask her for one. As a side note, if you've got the spy holodisk from the computer at Vault 15, tell Tandi about it and then Gunther for an additional xp and cash reward.
11. "Take care of Officer Jack" for Mira:
You'll probably want to take this quest on after you deal with Jack during the day. Read the section above the quests about Dorothy to complete step one. Next, you'll find Mira in the Bazaar area sporting green hair. In order to initiate this quest, you'll need a strong dumb PC, so if you don't have a high strength and low intelligence, take some drugs first to become so: mentats, buffout, psycho. She'll ask you to use the rope to climb over the wall to complete your business. Wait until nighttime, enter the downtown area and look on the eastern side of the map. Sneak into the police station and you can kill Jack in his cell. Return to Mira for your xp and weapon (if you offed him yourself).
12. Complete brahmin drive:
Ask Duppo for work, and he'll assign you the task of guarding the brahmin caravan on route to Redding. Meet with the caravan driver outside Westin's place to begin, and during your travels you'll be rewarded at the end an amount of money depending on how many brahmin survived the various encounters. Now you can walk back to NCR for your car if you have it and left it there. No xp for the quest but a little cash... probably not worth the time it takes if you're rushed.