Both abilities trigger simultaneously. Then, since they want to go on the stack at the same time and you control both triggers, you decide the order in which they go on the stack. They will then resolve in reverse order.
However, this order is quite irrelevant. The Flamespeaker's ability lets you play the exiled card for the rest of the turn, so you could simply let that trigger resolve first, exile a card, then let the untap trigger resolve, move to your second main phase and then play the card. In fact, you do not and cannot play the card when the Flamespeaker's trigger resolves. Instead you are bound by all the normal timing rules. I.e. if it exiles a sorcery, or a permanent card (without Flash), you will have to wait until the second main phase to cast it.
Lets say when his first strike damage hits I then untap and exile a lightning bolt since thos is a instant can I cast it then when the double strike hits untap the mountain I used?
Both abilities trigger simultaneously. Then, since they want to go on the stack at the same time and you control both triggers, you decide the order in which they go on the stack. They will then resolve in reverse order.
However, this order is quite irrelevant. The Flamespeaker's ability lets you play the exiled card for the rest of the turn, so you could simply let that trigger resolve first, exile a card, then let the untap trigger resolve, move to your second main phase and then play the card. In fact, you do not and cannot play the card when the Flamespeaker's trigger resolves. Instead you are bound by all the normal timing rules. I.e. if it exiles a sorcery, or a permanent card (without Flash), you will have to wait until the second main phase to cast it.