Regarding the first time that France went to war with Austria during the revolutionary period, this war with Austria was declared on April 20, 1792. It was the Legislative Assembly which declared war on Austria.
There were a number of reasons for this declaration. The revolutionaries hoped to spread their revolution. In addition, they hoped that going to war would ensure that domestic enemies would stop trying to bring down the revolution. They hoped the war would also bring the most revolutionary/radical of their leaders into power. Conversely, King Louis XVI and other monarchists thought that the war might help them, especially if the revolutionaries lost. They hoped that Austria and other countries might help to restore the monarchy.
The French Legislative Assembly declared war on Austria for revolutionary reasons. They wanted to spread the revolution and to protect and strengthen it at home.