Elemental (Fire/Ice/Elec/Wind) effects DO stack. However, the stacking is limited to a total of TWO skills or accessories, and skills take precedence over accessories.
(+) Fire Boost and Fire Amp
(+) Fire Boost and Fire Vow
(+) Fire Amp and Kagutsuchi Vow (Amp and +50% accessory is the largest effect you can have)
(+/-) Fire Boost, Fire Amp, and Kagutsuchi Vow results in only Fire Boost and Fire Amp stacking
Resist effects from affinities, skills, equipment, and spells stack:
(+) Ukobach (innate Resist Fire) + Resist Fire (learned at Level 6) + Kagutsuchi Pin (Fire damage down 50%) + Red Wall all stack
Barring Rules 1 and 2, a passive skill and an accessory providing the same effect NEVER stack.
(-) Apt Pupil and Omega Drive (quadruples critical hit chance? I wish)
(-) Angelic Grace and Cat Whisker
(-) Endure and Partial Award
Barring Rule 3, HP and SP regenerative effects DO stack and take effect in succession.
(+) Regenerate 1, Regenerate 2, Regenerate 3
(+) Regenerate 1, Regenerate 2, Space Badge
(+) Invigorate 1, Invigorate 2, Invigorate 3
(+) Mage's Mark, Invigorate 2, Invigorate 3
All other multi-tiered effects do NOT stack. The highest-tiered effect takes precedence.
(-) Growth 1, Growth 2, Growth 3 (only Growth 3 takes effect)
(-) Counter, Counterstrike, High Counter (only High Counter takes effect)
Ailment Boost and the Boost skills of individual ailments do NOT stack.
Ailment Boost takes precedence.
Other combinations that either do or do not stack:
(+) EXP-boosting weapon and EXP-boosting armor (i.e. Musashi Shinai and Haikara Shirt/Mandara Robe)
(+) Cool Breeze and Relaxing Wave
(+) Apt Pupil/Omega Drive and Revolution (thanks Nasada19)
(-) Evasion skills/equipment granting evasion skills and an affinity that is better than Resist (i.e. Null/Repel/Absorb). In other words, for example, if you have Null/Repel/Absorb Physical, you will never dodge physical attacks (meaning enemies will never miss and fall down). This is because the game assumes a general mechanic that prefers the automatic nullification of the attack. This may be something to think about if you really like letting enemies miss and fall down; in that case, something like Tam Lin's innate Resist Physical combined with the Evade Physical he learns might be more up your alley.