Imagine a subway map like the London Underground. It has lines branching off from stops and connecting them with little concern for actual geography. Many historians believe this is how ancient travelers viewed the world. Compare the Tube with the Peutinger Table, for example.

Anyways this means you will be traveling from town to town via roads and eventually you will end up in a town that is unmistakably Roman. You might begin to see villas with baths (giant complexes with pools and hot tubs) and towns with basillicas, amphitheaters, and fora (plural of forum). These are thoroughly Roman architectural forms and are used by archeologists today to study at which point Roman culture arrived in certain parts of the empire. The period of this journey was right when "Romanization" was at its peak.

The answer would be different depending on where you entered the empire. Gaul was pretty "romanized" at the time and the German border had a lot of barracks so you might have passed an army shortly after entering. Not to mention the Rhine was a definitive border at the time. In contrast you might have traveled quite a bit south in Britain before you saw the first indisputable evidence you were in Roman territory before Hadrian's Wall.

Romans' view of "borders" changes over time but in general they tended to think of the peoples and cities they controlled and protected rather than territory. Rivers like the Rhine and Danube did a pretty good job defining the northern borders but settlements on the opposite bank probably felt pretty connected to the empire rather than divided from it. On the south the desert was a convincing border but the eastern border was often disputed. This was shared with Parthia and Armenia and Rome often went to war with Parthia over issues in Armenia since they both viewed it as a sphere of influence. That region of the empire also had a stronger local culture and thus towns might not have Roman building types.

You would have a pretty good idea right at the border to be honest. The Romans did exercise border control with customs, tariffs and the like and the borders were also fortified. These were more "zones of militarization" than hard lines, but the forts, eagles, soldiers and people telling you in Latin to declare your property would be a quick tip off. And of course there were the roads. Even beyond that though the material Romanization of civilian areas was pretty extensive in most places by 117 CE. In Britain, for example, even though villas (which had started to spring up a few decades prior) often incorporated aspects of local significance like ritual sites their appearance was very distinctive from anything that came before.

There were definitely areas where Roman material culture found weak purchase, such as interior Sardinia, upland southern Aantolia and northern Wales but those weren't really next to the border.

Well, well, well, this is quite a contentious topic when it comes to scholars. I did my dissertation on the wars of Trajan and what his goals were (if he had any) and the issue of the empire's borders was one I dealt with. Admittedly I probably could have done more reading on it to better understand and discuss the topic but it was only a 10,000 word dissertation so I couldn't go into immense detail on this when I had to write about his wars in Dacia and Parthia (as well as Rome's history with them) and I managed to get a high 2:1 for it (that'd be close to an A in America, I suppose) so I must've done something right.

Historians like Luttwak have argued that the Romans had very clearly understood "strategic" borders that were methodically planned out and guarded. The Romans knew exactly where their empire ended and made sure to protect it adequately, with troops stationed along or behind the border to engage any advancing enemies. I believe Luttwak refers to the Romans as having a thin line of initial defenders backed by more concentrated units as "defence in depth."

Other historians have disagreed with Luttwak though; Susan P Mattern argued that the Romans had virtually no real idea of where their frontiers ran because they had terrible cartography and so wouldn't have known their exact borders as Luttwak implies. From what I remember she believed that the Romans certainly placed troops throughout their empire but that it wasn't really along a clearly marked border, more of kind of a thrust out into "no man's land" where they could be behind or beyond their true "border." She points out that while ancient sources praise the Romans as guarded their empire's frontier she sees this as non-military personnel glorifying Rome's military achievements, thus being biased and not really knowing what they're talking about.

I'd imagine it'd be fairly easy to cross in and out of empire provided you weren't near one of their roads or border forts - and considering these were in specific places that means there'd be a long stretch of the frontier that, for practical reasons, wouldn't see much if any Roman military presence. In this respect I'd imagine you could quite comfortably just keep walking through the country until you reach the first town, military outpost or Roman road.