How were people able to chart coastlines, islands, currents (etc) without airplanes and satellites?
This is, although it might not seem like it, effectively two questions; how did people navigate without airplanes and satellites, and how did they make maps without them. The reason for this is that maps are themselves just forms of data visualisation, and simply become one potential part of a navigation process rather than being automatic or required. One can potentially navigate coastlines, islands, and currents without a map being involved at all.
Fair warning, I can't talk about the sum history of all maritime navigation ever practiced, or of all cartography, because that's a huge subject that has specialist study in its own right. What I'm familiar with is the premodern history of navigation in the Mediterranean, which I still think gets us a pretty decent way to answering the questions at hand.
It is tempting to assume that all navigation and orientation prior to the use of cartography was simply rote learned, with all new navigational knowledge acquired by sheer brute force. But this overly privileges the concept of a drawn map I think, and understates the rigour and science by which new navigational knowledge was acquired prior to the development of 'precision' instruments. There are a lot of frustrating gaps in our understanding of how early navigation developed, leaving us simply to wonder at the ingenuity of history's nameless pioneers. But we know that these navigational feats were accomplished, and we are aware of the ancient Mediterranean texts known as peripli where an author would describe the relative locations of important ports and coastal landmarks and the distance between them. The earliest attributed examples are I believe Phoenician/Carthaginian but one must entertain the possibility that this simply reflects the earliest known written ones, and that this ultimately reflects oral knowledge of a far older pedigree. The most famous of these is arguably the Periplus of the Erythraean Sea, which describes key locations on the Red Sea coast of Africa, Horn of Africa, and the western coast of India, along with the distance between them and the way one utilises the monsoon winds to cross the Indian Ocean at the correct time of year. We are aware of far more of these peripli that only survive as attributions or quotes in other works, but this iceberg effect implies a widespread creation and proliferation of these texts. It suggests the deliberate distribution of navigational knowledge, and that a great deal of such knowledge was in circulation, and not simply within the Mediterranean.
However, these texts do not seek to systematically catalogue every inch of coastline, much as Roman itineraria do not attempt to catalogue every square mile of land in Europe (these Roman texts focus on the same notion of relative distance but between cities on land rather than by water). These are both forms of point to point navigation, where one is aware of relative direction, distance, and landmarks along the way, along with the general presence of something hazardous in an area. By these means a traveller can reasonably navigate between key locations with certainty without, at any point, carrying a picture in their minds that necessarily resembles a modern 'correct' geographical map. There is a hierarchy of useful information in which cataloguing every inch to instrumental perfection is not really a concern. In modern terms you can call it a matter of only using the underlying data that one finds useful, and in the case of a periplus or an interarium.
The oldest known examples of cartography, found in Mesopotamia and Greece are not examples of precision charting either. They are as much a matter of representing balanced division of the known world as they are in any sort of navigation, and some abandon the latter entirely. In other words, they're often diagrammatic rather than representative. The first attributed 'map', made by Anaximander, has been reconstructed as creating a relatively simplistic arrangement of Europe, Asia, and Africa (referred to on the map as Libya) in which all are roughly equally sized, all are oriented around the Mediterranean almost like wedges in a circle, and are all encircled around their outer edges by the same ocean. This remains a common style and genre of map well beyond these ancient eras and cultures, with most medieval mappa mundi eschewing any notion of representing actual relative distance and placement at all. This is another reminder that we have to separate the idea of accurate navigational/geospatial data and cartography, only because one doesn't require a map to navigate but also maps were not often made to follow the kind of accuracy you're imagining.
We're also aware of maps that exist as visual representations of Roman intineraria, and retain their focus on relative distance between cities but not an accurate relative position. These are mostly known from the medieval era, much like the mappa mundi. The Peutinger Table is arguably the most famous of these. It was made in the 13th century but there are... arguments, shall we say, about whether it's a copy of a Roman original or something later on. What (so far as I'm aware) isn't under dispute is that it's representing information from the Roman era, both in terms of relative distance between cities but also the route of Roman roads and important locations on them. This remained important information, and we know that pilgrims highly valued these sorts of maps- Mathew Paris, at the start of his 13th century work Chronica Major, has a seven page map showing the important stops and cities between London, Rome, and Jerusalem.
The first maps that begin to resemble our notion of geospatial precision are arguably the Portolan maps produced from the 14th century onwards. These begin somewhat similarly to a Greek periplus by being primarily concerned with ports and their relative distance, but unlike a Periplus or other maps there is a relative scale, and thus an attempt at showing the specific shape, relative position, and size of islands or of other coastal features. They use latitude, longitude, and rhumblines/windroses as key reference points, which is part of how they keep the maps to scale. In the case of the latter this involves accurately recording relative bearings so that those using the map can know what destination one would reach if one kept a consistent bearing. These are still, ultimately, collections of observational data from experience sailors and mariners, simply schematised to a high degree, and again underline my somewhat repetitive point that this data is not dependent on the map, the map is an intuitive visualisation of the data someone else possesses. It enables one to benefit from their experience without requiring their direct tutelage, or one benefits from the cumulative experience of multiple experience mariners at once, but the 'data' exists without that specific diagram being made.
Beyond this we get into the history of cartography in the modern era, which I am familiar with by trade but not historically, at least not to the point where I'm comfortable talking about it with as much surefootedness. But I feel like even these older examples have illustrated the point that cartography is not necessarily key to navigation at all, it simply helps proliferate navigational knowledge and schematises it. Nor has a lot of cartography ever really been about charting anything geospatial. The fact that you've linked cartography and navigational knowledge to airplanes and satellites show that you probably instinctively connect maps with perfect geospatial accuracy from a birds eye, impartial view. The history of premodern navigation shows that this isn't needed in order to get where one needs to be going. Humans are pretty good at collecting, remembering, and comparing observational knowledge, even without having the ability to look on the Earth from aloft.
On a sunny day, all you need is a protractor and plumb bob to check the height of the noonday sun and figure out your latitude. At night in the Northern Hemisphere, the North Star gives even more precision. Longitude was much harder, and the invention of accurate clocks was the eventual (most popular) solution. I sometimes explain it to grade-schoolers like this: if you set your clock to London time when you sailed westward, a few weeks later you would notice that the sun was directly overhead when your clock said 6 pm. You're a quarter-day off, and that means you are a quarter of the way around the earth, or 90ºW.
So if you have a goodly number of latitude-longitude observations for various points along a shoreline, you can sketch the parts in between quite accurately—supplemented by taking bearings of the coastline (for instance, noting that a particular mountain or headlands is due east of you when you're at a known point). Coastlines, of course, were visited frequently by folks (sailors) who could move about easily, could see long distances, had the skills to record latitude & longitude—and who had a great interest in knowing exactly where they were. A 19th century atlas will show the Iberian peninsula almost as accurately as a satellite image.
Mapping of interior areas was substantially more difficult. You can use similar readings of latitude-longitude, as Lewis & Clark did to some extent, but it's slow, tedious work. Most early maps of continental interiors were instead based on a form of dead reckoning, with the explorers roughly estimating distances based on number of hours of travel, and recording observations about large bends in river systems based on compass readings or sun position. These sketch maps sometimes proved to have errors of more than a hundred miles when more accurate positions could be recorded.
For large-scale mapping of interior lands, you can also use triangulation to very accurately create a network of known points scattered across an entire nation, and this was well under way in some European nations by the 19th century—but triangulation is really slow and exacting work. From those known points, typically mountain peaks or other things visible from a distance, you can use simple compass bearings to fill in the spaces in between with a little less accuracy but much greater speed. In much of North America, the rectangular land survey system that divided the advancing frontier into townships and sections of farmland quickly filled in a lot of the blank spaces. This was done with a little less accuracy than precise triangulation, but was more comprehensive, because the surveyors recorded on their plats the rivers, forests, swamps, lakes, and roads they encountered while walking the section lines. Here, for instance, is the surveyor's plat of a portion of the Chicago area.
Much of the populated Western world was surprisingly well mapped by 1900, and even at large scales, mapped features typically are within 10 meters of the positions that could be more exactly revealed in the 1930s by widespread use of aerial photography.
Some previous discussions: