I have read several times that the Roman empire was akin to a protection racket. They would come with their armies, extract tribute, and more or less leave you alone to govern yourself. I would imagine that one of the governor's duties was to extract said tribute and protect the province from other cartels. But is there more to it than that? What would, say an average provincial citizen in Gaul experience before, during, and after Roman occupation? Would the poor provincials consider themselves roman?
This is a large question, especially considering that Roman administration of the provinces was not uniform across time or space. In many ways life for people living under Rome continued to be very separate from Rome itself, but the administration of the provinces was often much more hands-on than the characterization in your question would suggest.
Yes, a lot of Rome’s interaction with the provinces was about exacting tribute, and that had a significant impact on people’s lives because it reshaped local economies. In many parts of the empire, longstanding industries increased greatly in scale and became largely devoted to producing goods for Rome. We see this, for example, with Iberian mines and Egyptian farms.
Tribute to Rome could also take the form of soldiers for the military, thus many people from the provinces served in the Roman army. Relatedly, if you lived under the Roman empire there would be Romans in your region, sometimes in your city, mostly members of the military.
Rome forcibly relocated many communities. In an extreme example, Strabo (7.5.6) writes that at at least 6 different tribes in the Balkans were reduced to poverty or death by being forced to live on infertile land where they couldn’t farm enough food.
In areas that were already organized into states, Rome became involved in the existing governing bodies. We can look, for example, at Roman Egypt and recognize the same government institutions from Ptolemaic times, but with legal and regulatory changes instituted by the Roman administration.
It’s also worth remembering how violent the incorporation into Roman rule often was. Historic cities like Carthage and Corinth were wiped out; Caesar carried out deliberate genocides of entire tribes in Gaul (see De Bello Gallico, in particular the fate of the Eburones). Continued conquest brought not only mass death but also mass enslavement.
You also ask how much people living outside Rome considered themselves Roman. Legally, Roman citizenship was initially very geographically restricted but gradually expanded until in 212 CE it included all non-enslaved people living under Roman rule. In culture, regional and local non-Roman practices continued, but in many areas Roman practices were also adopted (see for example the flourishing Latin literary culture in Africa, including among non-native Latin speakers). Some people and some communities seem to have seen Romanness as part of their own identities, while others didn't. In inscriptions, we continue to see people make note of their ethnic identities in many parts of the empire hundreds of years into Roman occupation, but displaying an ethnic identity wasn’t incompatible with displaying one’s (cultural, legal, or other) Romanness. We might look at the example of the Batavi, a Germanic tribe; their ethnic identity as Batavi seems to have been quite important to them (they explicitly mention it in inscriptions a lot), and their early alliance with Rome and reputation as good soldiers actually seem to have been a big part of that identity and what made it important.
(Like half the articles I've read this year could fit as sources for this question, so let me know if there's a particular thing you want sources for and I'll dig something up.)