Luck affects the amount/quality of items dropped from enemies as well as (assumedly) your critical chance.
Strength vs Magic is pretty much a preference. John isn't wrong about Strength Demons being relatively easy to find, but then neither is fusing some good Magical Fiends (Daisouju can have pretty much every possible good debuff you need and Red Rider/Harlot are fantastic casters). Daisouju can be fused to have: Makatora (infinite MP to the group), Makakaja, Fog Breath, War Cry (to be replaced with debilitate later), Dekunda and Dekaja. Oh, and it gets Prayer too, which is a full heal for the party that also removes all status effects.
http://www.gamefaqs.com/boards/582958-shin-megami-tensei-nocturne/49457472?page=1 (Scroll down to Master LL's post for the formula).
There are also a handful of abilities that only the player can get that are considered some of the best in the game, and they are for a strength based character; although you can technically max out your stats either by grinding or by exploiting the lucky ticket (spend at least 1k and get 1 ticket, after 10 you get to open a mysterious box which can hold an incense).
As far as what stats you should be giving priority though, agi and luck should be at the bottom of the list, with Str/Mag (whatever you decide) being first. You do need some Vit, but it's only slightly more important if you decide to go Str, because Str has a lot of skills that either eat your health to deal damage, or their max damage is based upon your max health. Other than those skills though, Vit doesn't really need to be focused on until the later half of the game. I think My character sat at around 8 vit (only increasing via magatama) and never worried about dying except against expel/death attacks if I wasn't prepared for them. My character was a Str based character, but I had Tornado as well as Force Boost on him for a loooooong time, and after I got the Daisouju, pretty much every non boss fight ended as soon as the Demi-Fiend had his turn. So even if you go Str, you will want to have at least one area of effect damage spell (Can't really beat Tornado until much later on).
Agility is to help you dodge and to bump up your turn in the group order, but in general it doesn't matter. You get more dodge out of a group buff or enemy debuff, so it really only affects your order. You will however want to raise your Str/Mag/Luck to at least 15 by the time you are about two-thirds of the way through due to it being a requirement for something.
