The empire was pretty exploitative. Rome took what it needed and forced the provinces to manage with what was left. Each province would have contributed in a unique way and they were far from equal. Egypt, for example was the breadbasket (though Sicily and Africa were also quite fertile). Italy, Spain and Greece produced mostly olives and grapes (for wine and oil). The Near East was known as a gateway for fine goods like silk and perfumes.

Despite the unfair arrangement, everybody ended up benefiting just from the ability to trade so easily around the entire Mediterranean. That allowed each province to specialize based on its own resources. We see a lot of public building in the 2nd and early 3rd century that shows both the new wealth and also the spread of cultural influence from Rome itself. That's when the system reached maturity, but after the crisis of the 3rd century specialization became a liability and the Mediterranean economy never really recovered.

In terms of military contributions that changes over time. In the Republic commanders could impose taxes and levies in the province if they felt necessary and mostly just to support the military. This led eventually to civil wars where Roman generals could levy armies abroad to fight each other, so Augustus changed the system. He changed to a voluntary army in which people from all around the empire could volunteer, but they would be sent to a far corner. A Syrian might be sent to Scotland and a Scot to North Africa. This can be seen in mile markers which were labeled with information of the legions that made them.