Does first strike damage occur before normal damage?
Yup, that's one of the reasons Drana's powerful - she can pump up your other attackers in time to help them deal more damage. The point of first strike is that it happens first!
Here's how combat plays out, assuming you attack with Drana, your opponent doesn't block her, and nothing weird happens:

First Combat Damage Step

Drana and all other creatures with First Strike or Double Strike deal damage. Drana will deal 2 damage to your opponent, unless you've pumped her up somehow.
Drana's ability triggers, then presumably resolves, placing a +1/+1 counter on each of your attacking creatures, including Drana.


Second (&quot;regular&quot;) Combat Damage Step

All of your creatures without First Strike (including those with Double Strike) deal damage equal to their power, including the +1 from the counters they just got.



There are rulings on the Gatherer page for Drana to this effect:

Drana’s last ability triggers after combat damage has been dealt, so only attacking creatures that survive the combat damage step in which Drana deals combat damage will have +1/+1 counters put on them.
The +1/+1 counter won’t change how much damage Drana or any other attacking creature with first strike or double strike deals during that combat damage step. However, the +1/+1 counters that are put on attacking creatures without first strike will affect the damage those creatures deal during the regular combat damage step.