How does stacking and resolving work exactly?
Twincast copies Lightning Bolt, dealing 3 damage to whatever target you choose, then the original Lightning Bolt resolves as usual - a total of 6 damage. The first sentence of your question pretty much explains why. To be specific (and correct the detail about priority):

You cast Lightning Bolt, putting it onto the stack.
You gain priority before your opponent does. (Note that by default, it's assumed that you pass it - if you want to hang onto it, you should say so explicitly.)
You cast Twincast, targeting Lightning Bolt. (The stack now has Lightning Bolt and Twincast on it.)
You and your opponent both pass priority.
Twincast resolves. It creates a copy of Lightning Bolt, optionally with a new target, and puts it on the stack. (The stack now has the original Lightning Bolt and the copy on it. Just one copy - not sure why you asked if it'd copy it twice.)
You and your opponent both pass priority.
The copy resolves, dealing 3 damage to whatever target you chose.
You and your opponent both pass priority.
The original resolves, dealing 3 damage.

For the record, the relevant rules about who gets priority when:

116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives priority afterward.

This is slightly important in this case because it means that you have to cast Twincast before you see if your opponent wants to respond to Lightning Bolt; if you pass priority to them before casting Twincast and they do nothing, then you've both passed and Lightning Bolt will resolve.