So far I find toughness, extra damage for X weapon type (I use non auto rifles myself) because they give damage and defense bonuses without the need for resource gathering. The barter perk seems to give a nice 20% discount on prices and of course lockpicking and hacking. After I get those I move on to the crafting perks like armorer and gun nut.

Which perks do you find invaluable in beginning of the game?

The first two I pick will always be Lockpicking and Hacking.

Lone Wanderer if you go solo like myself, whatever the perk is that restores more health from Stimpaks, weapon damage upgrade, the ninja perk is nice early on, too.

Same here. The next few I chose were Scrapper, Gun Nut and Armourer.

I always start with a damage perk at first levelup, then the looting perks, then survival perks.

Awareness, then the weapon perk of your choice, locksmith, sneak.

Actually, the first perk was the one to breathe underwater and take no rad from water... Helped a lot more than you think. The others I choose tends to be lockpicking and damage related.

I got a rank or two of Lead Belly and Medic to begin with...then kinda dabbled based on what looked interesting.

Armor and weaponsmithing

Hacking because there may be some interesting stuff on there, and gets loot to boot

After settling on combat style then more into specific weapon types

Lone Wanderer, Scrounger, and Critical Banker.

GunNut and Armorer are the 2 first ones I invested in.