Fallout – repeat the quests Quartermastery and Cleansing the Commonwealth forever

Yes, using Mentats is just like having two extra Intelligence points for extra experience when handing in a quest. However, you can't stack more than one Mentats effect at a time (two extra points is the max).

Apparel

If someone could confirm this in the comments below, I'd appreciate it. I think the easiest place to find the hat+body items is the Quincy Ruins. You can head there at the beginning of the game straight past Diamond City. In the middle of the town (held by raiders) is a storefront (Guns, guns, guns, it's called) that has a bunch of clothing items on an open shelf. One of them is the Ushanka Hat. I also went to the top floor of this building and found a Vault-Tec Lab Coat in the dresser next to the bed. (Can someone confirm whether this is random or a set item?)

• Vault-Tec Lab Coat (+2 INT – locations)
• Press Hat (+1 INT – trade with Piper when she becomes your companion)
  • Alternative: Ushanka Hat (+1 INT – Quincy Ruins, see above)
• Liam's Glasses (+2 INT – at end of Plugging a Leak quest late in the game)
  • Alternative: Road Goggles (+1 INT – locations)

These items add to the Intelligence stat for extra exp gain atop Mentats when handing in quests. 10 (maximum base INT) +5 (apparel additional INT) +2 (temporary Mentats buff) = 17 INT points!

Sleep

Waking from sleep with a "well rested" notification denotes a 10% experience buff for 12 hours (though I haven't been able to confirm this myself). If you sleep with a romanced partner, the bonus increases to 15% ("Lover's Embrace").

"Institutionalized" is a main story quest and cannot be skipped but in order to get the achievement you must follow the Institute quest line instead of any of the other factions.

If you choose another faction to enter the Institute during the mission The Molecular Level, you'll end up doing part of the Institutionalized questline in many instances (for example, the Railroad quests have you continue the Institute quests concurrently with their own to avoid arousing suspicion of the Institute) but you'll not complete the required mission to get the Institutionalized achievement.