Shotguns are fun to play, but a) you want some back up weapon because they have short range, and b) you won't start out with one, so at level 1 you will be fighting with a pistol. I'd suggest normal difficulty with oddity experience. A pistol can work as back up (no str requirements, so if a melee character somehow manages to drain your strength making shotguns unusable, you can switch).
I don't like tanks, so this is for a light-armored, stealthy, mobile shotgunner:
strength 5 - this is the lowest you need for shotguns. I'd recommend putting the first two level up points here and ending it at 7 strength on level 8, for the full auto feat.
dexterity 6 - for the grenadier feat,
agility 6 - for the sprint feat,
constitution 5 - just for hitpoints (you could lower this to 3 if you like a more risky playstyle),
perception 10
inteligence 5 - after strength, this is where I'd put the next two level up points, ending with 7 intelligence at level 16 for crafting feats.
The last two level up points I'd put into perception, ending with: str 7, dex 6, agi 6, con 5, perc 12, will 3 and int 7 at level 24. Which isn't the max level, but you stop getting level up stat points here.
Starting feats: sprint, nimble, level 2: sixth shell (one of the best shotgun feats, worth specializing in once you start getting specialization points), level 4: grenadier. That should get you going, you can test out different build ideas here:
http://underrail.info.tm/build/Skills: guns, throwing, stealth, hacking, lockpicking, crafting, mercantile. You don't need to invest in crafting immediately, though, because you will have to find good components first. But in general what you can make will be far better than what you can find or buy, and some items are available only through crafting.
Psi - is it worth it? Yes, but you can do without it. You will want 55 points in temporal (it's the invested, not the effective skill that counts, so don't worry that your 3 will lowers it) and the two abilities: limited temporal increment (you can buy it at the starting location) and psycho-temporal contraction (this one you can buy only later in the game). LTI helps a lot with grenades, so it's useful early on. I'd also recommend 45 in psychokinesis for the following abilities: electrokinesis, force field, electrokinetic imprint. Electrokinesis & imprint stun living beings and short-circuit robots, force field provides a quick cover or a way to stop enemies from getting to you. Psycho-temporal contraction is basically a "haste" abiltity, which stacks with adrenaline.
But, as mentioned, you can do without psi if you don't like the idea.
One note on shotgun ammo: for lightly armored enemies 20p (hopshot) is best. If you find the enemies resist too much damage, work yourself up > 12p (houndshot) > 6p (boarshot) > 3p (tungsten trainshot). But in practice, I've found tungsten only rarely useful (and it weights a lot), because most of the time 6p did the job against resistant enemies.
edit: Since you won't have access to a shotgun at low levels, make use of whatever tools are available, e.g. bear traps require zero skill to set, molotov cocktails are very useful and remain so throughout the game.