Here's some of my favorite perks so far:
Rooted: Very useful for melee characters. The extra damage is nice, but the damage resistance is especially important as you cleave baddies in front of you that are firing with automatic weapons. A few points in this and I'm laughing as raider bullets bounce off my skin.
Rifleman: Very useful for the Snipers out there, although the same can be said to any of the damage enhance skills, like Big Leagues/Iron Fist (The 5th level of Big Leagues with the Grand Slam option is hilarious). I like Rifleman though, because Sniper rifles already have high damage, so you see nice gains from this skill. The ignored armor bonus is pretty nice against dudes in Power Armor.
I find I don't use VATs much with my Sniper, but when I do the Sniper/Concentrated Fire perks are pretty nice. I find it's best to save Action Points at the start of a fight though, taking head shots from a distance and then falling back to VATs if things get hairy.
The 3rd level of Lifegiver is good, if only for the auto health regeneration which you can get as early as 20.
The 2nd level of Aquaboy is fairly worthless, but the first level allows you to breathe underwather indefinitely and you take no radiation when walking/swimming. Pretty damn handy for just 1 point.
Solar Powered: the stat bonuses are nice, but level 2 allows the Sun to remove radiation, and level 3 regenerates lost health while in the Sun. Free healing is always nice when you're far from a bed, especially on Survival mode.
Local Leader: Pretty much a must have if you plan on doing the Minutemen settlement quests. If not, skip it.
Inspirational: The first two levels are handy, the 3rd level is helpful but not necessary.
Gun Nut: Even if you're a melee character, Gun Nut can be nice for building Turrets in your Settlements (Not required for the basic Turret though) and for building guns for companions. Keep in mind that you may need Science for some of these two.
Nuclear Physicist: Handy for the extra Fusion Core length, but the extra damage for Radiation weapons is particularly nice for just 1 point, and being able to eject Fusion Cores from other Power Armors can shorten fights if you can make it happen.
Nerd Rage: Hulk SMASH! Haven't gotten it yet, but looks to be overpowered when it procs.
Ninja: A must if you're a Stealth character. One shot Stealth kills are sweet.
Scrounger: Unlike previous games, everyone I've talked to has found this perk to be very useful in Fallout 4. I just got it recently, so I can't say one way or the other how I feel personally.
Idiot Savant: Everyone seems to like this one, although the sound effect when it procs can get annoying after awhile (Someone already has a mod out to remove the sound effect if you're on PC).
Grim Reaper's Spirit: If you use VATs a lot, this one is a nice perk to have all 3 levels in. Pointless if you don't use VATs though.
Ricochet: Another one I don't have yet, but 1-2 points in it looks really good from what I've seen. Downside is you need to be low on health to really see it come up often.
Scrapper: If you're picking up everything then this perk won't be as useful, but the extra Copper alone is helpful for Settlement building cause you go through a damn lot of Copper.
Those are just some of the ones I've used personally, or they're perks friends have tested. Plenty of other great perks out there. Some are no brainers, like Armorer, that's almost a given though. If you're Melee you'll obviously want Blacksmith for the same reasons as Gun Nut was listed, and it might be good to have anyway if you run with a character that prefers melee, like Cait.
