SPEECH: There are a LOT of quests that can be skipped entirely with Speech and several things that can't be initiated without a high Speech score (like recruiting certain companions). I like keeping this high, but I'd be curious to play without points in it.
BARTER: Most useful if you want the Pack Rat perk. I think most dialog options that require it for quest purposes only require it pretty low, so you can get by with a magazine and some Charisma-boosting booze. Otherwise, it's not as if caps are at a shortage.
SCIENCE: I think you can get by with only a few points in Science (unless you're playing Energy Weapons or want to be able to complete the one or two "Science 90" dialog options). Most terminals with quest-crucial info are Average or easier, and between clothing, magazines, and chems, that means science can be as low as 26 (or 16 with Comprehension).
MEDICINE: Similarly, Medicine can safely be pretty low, as most medicine speech checks can be passed with a magazine and/or clothing with your default stat if your Intelligence is over 5. I think there's only one optional quest that requires Medicine to be pretty high, and you can still complete the larger quest it belongs to without doing that part.
LOCKPICK: Handy to get some special items, and reassuring yourself that you have seen all you can see, but I don't think it's quest-essential in any case.
SURVIVAL: I think is fine to leave at its default, or 45 if you want Travel Light and/or Animal Friend. If you are playing a Melee/Unarmed build, though, Rad Child is one of the best perks in the game. Which brings us to....
What's with the Radchild love?
Critical Radiation Poisoning put an -3 END, -2 AGL, -1 STR debuff on your character, and you still need 10 sec to cover the lost health from END -3.
I think you'd need to try playing with Rad Child to see how amazing the effect is. With DT-enhancing perks and implant, even in light armor, you can basically wander into any firefight and automatically heal all damage from gunfire. Keep in mind, too, that the health regen from Rad Child stacks with other healing effects in hardcoe. My strategy for a certain cave full of deathclaws was to eat a couple snacks, drink a Nuka Cola, and wander in with my fists raised. You heal up almost instantly after being struck. While it does mean a loss of max HP, it represents a constant gain, as if you've got a stimpack on IV.