For the radiation you'd be better off with electric or even fire, poison is not exactly applicable in that case.
I would make this:
Normal = normal, plain, anything not fully covered by the other types
WEA -
INM - ghost (GFs choice)
RES -
STR -
Grass = plants, life, fungus
WEA - fire (fire burns plants), bug (bugs eat plants), ice (ice kills plants)
RES - electric (easy grounding), grass (duh), water (plants feed on water)
STR - water, grass, rock = 3
Fire = fire, heat, sun
WEA - water, ground (dirt also can kill fires), flying (air can kill fires)
INM -
RES - grass (fire burns plants), fire, steel (metal melts at high temperatures), fairy (GFs chioce), bug (fire kills bugs)
STR - grass, bug, poison, steel = 4
Water = water, fish
WEA - electric (water conducts electricity), grass (plants feed on water), ice (cold freezes water/water conducts easily the cold)
INM -
RES - fire, steel (rust), water, ground (dirt and sand lose cohesion in water)
STR - fire, ice, ground = 3
Electric = electricity, light, radiation
WEA - ground, rock (similar composition as ground, also, rock is usually grounded)
INM -
RES - electric, steel
STR - water, ice, flying, ghost = 4
Ice = ice, cold
WEA - fire (fire melts ice), steel (harder than ice), water (water at regular temperature merges with ice and melts it), electric (good conductor)
INM -
RES - fighting (cold muscles perform worse), ice
STR - grass, water, flying, bug = 4
Flying = birds, air elemental, flying or soaring creatures
WEA - electric (ever seen a bird hit by lighning?), rock (two birds? one stone? get it?), ice
INM - ground
RES - bug (birds eat bugs), fighting (GFs choice)
STR - fire, bug, fighting = 3
Bug = insects, arachnids, all arthropods in general
WEA - fire, flying (birds eat bugs), poison (poison kills bugs), ice (extreme temperatures kill bugs)
INM -
RES - psychic (freaks them out, they lose focus), grass (bugs eat plants), ground (GFs choice)
STR - grass, psychic, dark = 3
Poison = poisonous creatures, contamination, corruption, harm, death
WEA - psychic (renders poison useless and prevents it's spread or whatever), fire (burns the noxious substances)
INM -
RES - poison (inmunity), fairy (corruption), grass (poison can be harmful to plants, but I don't see it enough to be worthy of a weakness), fighting (the physical strength and will are hampered)
STR - bug, fairy = 2
Psychic = the powers of the mind, kinetic abilities, precognition, space, both outer space and the dimension
WEA - dark (a common phobia), bug (common phobia), ghost (common phobia)
INM - fairy (the rational mind is blind to magic)
RES - fighting (brain beats brawn), normal (the normal against the extraordinary)
STR - poison, fighting, rock = 3
Fighting = physical strength and prowess, will, honor, justice
WEA - psychic (brain beats brawn), fairy (GFs choice; also, it's funny), flying (GFs choice)
INM -
INM -
RES - dark (trickery always loses to JUSTICE), rock (tough m********s, the fighting type), fighting (who knows better how to counter fighting than fighting?)
STR - rock, steel, dark = 3
Ground = earth, dirt, sand, mud, deserts
WEA - water (dirt loses cohesion in water), grass (feeds off the ground)
INM - electric (inmediate dispersion)
RES - rock (they are pretty much the same), ground, poison (kinda hard to poison the ground), fire (can the ground burn?)
STR - fire, electric = 2
Rock = rock, minerals, gemstones, mountains, terrain, bitchin' guitar shredding
WEA - steel (steel is harder than rock, usually), grass (same reasons as ground), psychic (rock is easy to break for someone that can manipulate matter), fighting (strength beats hardness)
INM -
RES - fire (fire can't normally burn rock), normal (regular ol' hardness), poison (same with ground, rock is inanimate),, flying (rock resists air and birds), bug (rocks are harder than ground for bugs to use), rock (rock is composed of similar minerals to the ground, and is often grounded)
STR - electric, flying = 2
Steel = metals, artificial, mechanical
WEA - fire (fire melts metals), electric (good conductor), fighting (GFs choice)
INM - poison (steel is impregnable by poison, also, many times it might not be exactly "alive", per se)
RES - normal, grass, ice, flying, bug, dark, dragon, fairy, steel
NEU - ground (GFs choice, nerfed), psychic (same reasoning as rock, but harder to pull off), fighting (GFs choice, nerfed), ghost (physical barriers and technology can't stop the supernatural), water (water rusts metal),
STR - ice, fairy = 2
Ghost = ghosts, supernatural, possessions, BOO!
WEA - ghost (ghosts can touch ghosts), electric (being beings of energy, energy can hit them), dark (GFs choices), ghostbusters! nanananana nananananana nananana nanananana who you gonna call? ghostbusters!
INM - normal (can't touch 'em), fighting (same reason)
RES -
STR - psychic, ghost = 2
Dark = darkness, night, trickery, evil, cruelty
WEA - fairy (good beats bad), fighting (honor beats trickery), bug (bugs are good in the dark)
INM - psychic (freaks them out, lose focus. also, it's hard to manipulate something you can't see)
RES -
STR - psychic, ghost = 2
Dragon = dragons, fierce and mythic creatures, legends, dinosaurs, lizards
WEA - dragon (dragon can beat dragon), ice (cold blooded), fairy (magic can beat dragons)
INM -
RES - grass (GFs choice), water (GFs choice), fire (GFs choice), flying (rounding up the whole elemental theme), ground (same reason)
STR - dragon = 1
Fairy = fairies, magic, non-fierce legends, supernatural not quite fitting for psychic and ghost
WEA - steel (something about fairies being weak to iron), poison (corruption)
INM - dragon (GFs choice)
RES - psychic (magic is not affected by rationality), dark (GFs choice), fighting (GFs choice)
STR - fighting, dragon = 2
And that's it. if you think something is unbalanced, tell me, I think it makes sense, modifying a few things GF came up with.