The damage roll is all the dice together. Add the bonus after.

When you deal damage with a single* distinct attack or other effect, you roll all the indicated damage dice, including any dice added by critical hits or special effects you benefit from like a curse, hex, or sneak attack. That counts as making the damage roll. If you have a feature that lets you add a bonus to the damage roll, that happens after totaling all the dice rolled.

The rules you seek are on page 196 of the Player's Handbook under the Damage Rolls heading. Here is the most relevant passage for this instance:

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.

In your example, you will roll two d8's and one d6, add the numbers on those three dice, then add your proficiency bonus (such as +2, for example). The result is the damage you deal.

If you have multiple bonuses due to some combination of effects, they are all processed after the dice are rolled collectively, unless the text for the effect indicates that you add the bonus to each die individually or some other exceptional process.

(*If you're actually making multiple distinct attacks or dealing multiple separate instances of damage such as with a spell with multiple targets damaged separately and not simultaneously, each has its own distinct damage roll. The rules for the attack, spell, or other effect will explain how and when damage is dealt. Basically, whenever the text tells you to deal damage involving dice to a particular target, that's a damage roll for that target. Read the rules under Making an Attack and Casting a Spell in chapters 9 and 10 of the PHB for how to deal with other cases.)

About 27%, or a little over 1/4

This is a pretty easy calculation to run in anydice:

As you can see, the percentage chance of rolling at least a 19 one of the d20 rolls is 27.10% So you can expect, on average, to crit a little over once in every four attacks made in this manner.

How do you calculate that without a simulation?

The probability of getting at least one 19 or 20 among the three dice is equal to 1 minus the probability of not getting any 19s or 20s among the three dice.

Since there is a 2 out of 20 chance to get a 19 or a 20 on each die, we know that there is an 18 out of 20 chance to not get any of those values. 18/20 = 0.9, so there is a 90% chance we won't get any 19s or 20s on any given die.

To find out the odds of not getting a 19 or a 20 on any of the three dice together, we just have to multiply the probabilities for each individual die together.

So, the probability of getting at least one 19 or 20 among the three dice would be $$1 - (0.9 \times 0.9 \times 0.9) = 1 - 0.729 = 0.271$$

Multiplying that probability by 100 gives us a 27.1% chance to crit on any given attack.