"I will bring my dog" is in the future tense and uses a modal verb, will. You can use other modal verbs: can, might, should, must, etc.
Notice the use of a comma (,) at the end of the first clause. The use of "if" creates a dependent clause. The order of the sentence may also be reversed:
I will bring my dog if I go to the park tomorrow.
Here are some more examples:
If Aishwarya goes to the party, she will wear this beautiful necklace.
*Notice the comma ( , )
after "party."
Pablo will be very happy if he can marry the girl he loves.
If I buy this house, I’m going to paint it a different color.
Click below for a description of the future conditional: