To easily gain experience, 30 Speech skill and neutral or friendly with NCR are required. Go to Camp McCarran and speak to Sergeant Bitter-Root in one of the tents. Ask him about his name, and an option will display for a 30 Speech skill attempt. You can keep saying "Bitter-Root. You have an unusual name.", and each time you say it, you can do the Speech attempt again. Simply say "Bitter-Root. You have an unusual name.", do the Speech attempt, ask to talk about something else, and then repeat as desired.
This trick requires 30 Speech skill or the "Confirmed Bachelor" perk and friendly with NCR. Go to HELIOS One, and a woman will talk to you. After she allows you access to HELIOS One, you will be able to start the "That Lucky Old Sun" mission. Proceed with the mission, and talk to Fantastic and Ignacio Rivas. While talking to Rivas, make sure you get him to reveal that he is a Follower of the Apocalypse by using Speech or Confirmed Bachelor, and agree with his ideals about peace or say you are neutral. Continue with the mission until you get to the point where you activate the Mainframe Terminal. Configure the Power Grid to "5. Full Region (Emergency Output Level)". Finish the mission by hitting the Reflector Control Panel outside on top of the tower. Go back to Ignacio Rivas, and choose the "I overloaded the plant. No one....." option. That option will never disappear. Keep selecting that option to get 350 XP, 3 Stimpacks, and 2 Doctor's Bags.
-After reaching a Speech skill of 50, you can persuade Old Ben to offer his escort services to the local bar. Old Ben is sitting by a fire close to The King headquarters in Freeside. You will get 61 XP each time you complete this Speech Challenge. If you follow him back to the bar afterwards and wait until he sits down, you can talk to him again and repeat the same Speech Challenge again as many times as desired.
-This trick requires a high Speech skill and 8 Intelligence. Go to the Camp McCarran Terminal building inside Camp McCarran. Talk to the guy at the far end of the building, in the left corner. He is the leader and will give you multiple quests. Accept the interrogation quest, then go meet with the woman upstairs to begin the quest. When you get in the room to talk to Sirus, pick the "Intelligence" option first, the "Speech" option next, and then the "Intelligence" option. Any of the three "Intelligence" options work. Then, select the "Speech" option, and cycle through the "Intelligence" options. You will gain 50 XP each time that you cycle through the options.
-This trick requires a Medicine skill of at least 35. Once your character gains access to the main strip, find Victor outside the Lucky 38 casino. He will tell you that Mr. House wants to meet you. Follow him inside and go up the elevator. Speak with Mr. House (the giant television face that looks like Howard Hughes). Find the dialogue choice pertaining to needing to ask him a few questions. Then, ask "Who exactly are you, Mr. House?", then select "You appear to be a computer, not a man". After that, select the Medicine skill check response, "The lifespan you're claiming is impossible, except for ghouls and super mutants." After that, you will be awarded 35 experience points (or 39 experience points if you have the Swift Learner perk). Repeat as desired.
Go to a casino and play roulette to always gain 100 experience points.