Funny, I was just reading up about this. Strabo finished the last version of his Geographica ca. 23 AD. Regarding Scandinavia and Eastern Europe, he writes:

Of the Germans, as I have said, those towards the north extend along the ocean; and beginning at the outlets of the Rhenus, they are known as far as the Albis; and of these the best known are the Sugambri and the Cimbri; but those parts of the country beyond the Albis that are near the ocean are wholly unknown to us. For of the men of earlier times I know of no one who has made this voyage along the coast to the eastern parts that extend as far as the mouth of the Caspian Sea; and the Romans have not yet advanced into the parts that are beyond the Albis; and likewise no one has made the journey by land either. However, it is clear from the "climata" and the parallel distances that if one travels longitudinally towards the east, one encounters the regions that are about the Borysthenes and that are to the north of the Pontus; but what is beyond Germany and what beyond the countries which are next after Germany — whether one should say the Bastarnae, as most writers suspect, or say that others lie in between, either the Iazyges, or the Roxolani, or certain other of the wagon-dwellers — it is not easy to say; nor yet whether they extend as far as the ocean along its entire length, or whether any part is uninhabitable by reason of the cold or other cause, or whether even a different race of people, succeeding the Germans, is situated between the sea and the eastern Germans. And this same ignorance prevails also in regard to the rest of the peoples that come next in order on the north; for I know neither the Bastarnae, nor the Sauromatae, nor, in a word, any of the peoples who dwell above the Pontus, nor how far distant they are from the Atlantic Sea, nor whether their countries border upon it.

Some notes to help put things into context, along with some of my own observations:

Strato used "Germans" to refer to all native inhabitants of Germania, which was a region much larger than modern Germany (map). So when he says the Germans extended along the ocean, he is talking about the Frisii, or Frisians.

The "Rhenus" and "Albis" are what we now call the Rhine and Elbe rivers. The Sugambri have disappeared from history. The Cimbri are identified both here in the Geographica and also by the historian Tacitus with the Cimbrian Peninsula, which is usually thought to refer to Jutland. But they are also mentioned in connection with the battle at Teutoburg Forest. So I would say that today's Schleswig-Holstein is a good candidate for the northernmost land area that Romans visited, since this is where different sources begin to disagree. Augustus would not let his generals chase defeated tribes when they fled over the Elbe river.

(In Rex Gestae Divi Augusti, Augustus himself mentions a fleet led up to "the frontiers of the Cimbrii", where it was forced to turn back due to fog and ice. Strabo does not mention this.)

From here, Strabo goes on to conclude that by sailing further east (i.e. into the Baltic sea) one would be able to reach the areas around the "Borysthenes", i.e. the Dnieper river in Ukraine. He's not too far off: one of the Dnieper's tributaries, the Berezina, extends as far as norther Belarus. So the area around what is now Minsk seems to me as a likely candidate for the most northeastern parts of Europe explored by Romans.

Strabo's world map (modern reconstruction) shows a smooth Northern Europe with no Scandinavia, and a combined Northern/Baltic sea that one can navigate all the way to the Caspian Sea. The entire landmass of Russia is missing. But Strabo did not rule out the possibility of an unexplored landmass, labeling the area "Terrae incognitae" (unknown lands).

A great book for further reading would be Space, Geography, and Politics in the Early Roman Empire by Claude Nicolet. Chapter 4 "Explorations and Expeditions during the Roman empire" is mostly available here on Google Books.

Adding to the answer /u/Beastybeast gives regarding north, I can say the farthest east individual Romans seem to ever show up is around Vietnam. There was a pretty bustling sea trade between the Roman Empire and India, Strabo giving the number of ships departing every year as 120, which would be pretty crazy (to give a kind of weird comparison, there were about 1500 ships sailing between England and colonial America, so imagine the ship traffic between Rome and India as almost a tenth that, which is really a whole lot). It is difficult to know whether Romans regularly traveled to the Bay of Bengal regularly, but they did on occasion, and apparently went even farther. The specific incident I mentioned was in 166 CE a Chinese garrison picked up a group who claimed to be an embassy from Rome--These were probably just merchants who claimed to be an embassy for safety reasons. A later chronicle also somewhat offhandedly says that Roman merchants were active in the area in the third century, but who knows how to handle that--they may have been Bactrians, they may have been merchants who dealt in Romans items, or indeed they may have been Roman merchants. Regardless of how you take these, I would say they constitute a pretty impressive distance eastwards.