Poison: Loses effectiveness against grass. Gains effectiveness against fairy and water.
Fighting: Loses effectiveness against ice.
Ground: Loses effectiveness against steel.
Rock: Gains resistance to rock and electric.
Flying: Loses effectiveness against grass. Gains effectiveness against ground.
Steel: Gains effectiveness against fairy. Gains weakness to electric (and/or water).
Grass: Gains effectiveness against fairy. Loses ineffectiveness against steel and poison
Ice: Gains immunity to ice and resistance to water.
Poison would be able to serve a much better offense role hitting water and fairy, both of which can be common and perhaps have few other exploitable weaknesses, losing its less useful effectiveness against grass (and bug also being unneeded). Fighting and ground would be nerfed a bit, which in turn would help to bring other types into prominence (steel would become more viable against ice, bug would become more viable against grass and dark, fire becomes a bit better). Flying, by being more distinguished in its effectiveness array from fire, would become more offensively viable. Grass would be less of a butt monkey. Ice and rock would gain defensive niches.
After plugging these into the type chart & mulling it over a bit, I'm actually rather fond of these suggestions. It seems like it would do well to bring some of the lower-tier types out of the shadows. I especially like the idea of grass losing the flying/poison weakness, but that's partly because it's my favorite type~ The other part is that poison's effectiveness to grass is pretty much pointless considering that right now ice, fire, & flying are much better checks. I agree that even if poison is only effective against two types, by handing poison weakness to fairy & shifting grass' weakness over to water, poison gets both a more solid offensive boost & a little breathing room.
I'm curious as to what fairy's other battle properties are...
At the very least I'm hoping for effectiveness against ghost, rather than dark, and ineffectiveness against Ice, which would be more beneficial for ice-type pokemon as ice being super effective to fairy is really only going to benefit all of the water-types packing ice moves.
As far fairy's weaknesses go, I really just want poison/steel/grass, like what Aves suggested above. Steel & grass make plenty of sense lore-wise (I think I also suggested a weakness to grass many pages ago, aha).