This only applied to Speed and Slowness potions, Mining Fatigue and Mining Haste. Does applying on then the other (to yourself) change the way the effect you, or change the way the particles show? Also, how does applying them both effect your speed/mining rate. Do they cancel each other out to leave you with the rate you started with, or do they end with a slight buff/debuff?
Speed and Slowness
The Speed effect increases your player's speed by 20% per level. The Slowness effect decreases your player's speed by 15% per level.
Therefore, if you have the Speed effect with a level of
3*k and the Slowness effect with a level of
k is an integer) your speed will be normal.
Haste and Mining Fatigue
Haste increases your mining speed by 20% per level. However, Mining Fatigue is bit weird.
For level 1, your mining speed is 30% of its original (70% speed reduction). For level 2, it's 9% of the original, for level 3, it's 0.27%, and for levels 4 and above, it's 0.081%.
So there's no possible combination of of Haste and Mining Fatigue which would "cancel each other out".
Both colours of particles are emitted by the player in each of the above cases. Applying one effect does not remove the particles of the previous one.