The College of Lumina sent several explorers through magical gates they found in the ancient caves. They learned that these gates go to their moon. Now, the problem is, they have no way to get the explorers back and there is no way for the explorers to communicate with Lumina.
So how would this small colony of 20 explorers on the moon communicate with its home planet, without help of radio waves, satellites or modern technologies.
Assumptions...
You don't mention what level of technology your civilization has. Judging strictly from the lack of radio compared to Earth's tech tree, your civilization's tech is no later than Earth's late 1800s.
Granted, it's your world, but it's a bit odd that magical gates would take one to the moon but not back. The idea has a Barsoom feel to it (or maybe Escape from New York), but I wonder why anyone would create a one-way gate without a purpose. But, to be fair, you may have simply not explained why. You're not obligated to.
I give you... Morse Code
Humanity has been using various forms of light-based communication for a long time. I include in this things like semaphores that must be seen to be useful. However, running up flags and such won't buy you anything because moons tend to be a long honking way away. Even today we can't see the equipment left behind from the Apollo moon missions with Earth-based telescopes.
But what would be believable is light. By the late 1800s humanity knew pretty much exactly what the distance between the Earth and its moon was. This means it's possible to create a parabolic mirror that would focus the light for easier sighting from the surface of your planet by telescope.
After that it's just a matter of using something like Morse Code to have the same basic capacity as a telegraph used commonly at the same time period.
Yeah, but how'd they know to take the mirror?
Ignoring the fact that British Army officer Robert Baden-Powell held the first unofficial Boy Scout camp in 1907 and published the first book about scouting in 1908 (you know, Be Prepared!), the simple truth is that mirrors have been used to signal people for a long time. Archimedes used them in 212BC during the battle of Syracuse. The ancient Greeks are thought to have used the concept as far back as 490BC during the battle of Marathon.
The point here is that signaling mirrors would be believable and, factually, given that the signaler is using one correctly and the receiver is looking for it properly, it factually could work. But why would any intrepid explorer have a parabolic mirror in the first place?
Warning! Story nonsense ahead!
It's very believable that a reasonably well thought-out expedition would include basic mirrors. As mentioned, they'd been used to signal for millennia. Without radio, there would be perceived a need for reconnaissance teams to communicate with each other and the main encampment. So mirrors as a concept is pretty natural.
Then time passes and nobody returns to the planet. Worried leaders start wondering what's happened. You could rely on good luck leading someone to think to themselves, "hey, they have mirrors! They could be trying to signal us!" and so look up. Possible, but perhaps not the most entertaining way to solve the problem.
Another solution (and a reasonably believable one) is that the teams could have left with the understanding that they were expected to report back by some specified time. When everyone comes to realize there's a problem (the teams that they can't return, the planet that they haven't reported in on time), a discussion on the planet sends someone through who's very well armed just in case there's hostiles present.
What are you blokes doin' here? You forget to send someone home or somefin?
But that doesn't work, either.
Well, mate, hate to break it to yah... here, try a bit of this lunar rat, it'll make the ugly truth go down easier.
So an intrepid scientist (must be a scientist, the military isn't known for entertaining humanitarian solutions like, "hey, maybe the portal is one-way!" They're much more adept at "there's a four-headed beast on the other side that's eating all our people! Send a bomb!") comes up with the idea of creating a predictable signal.
Here's where you can have some fun with the science. The relationship between the star, planet and moon are not always conducive to light-based signaling. In fact, they'd be very rarely conducive to a two-way conversation (hold that thought...). There needs to be either sufficient light to signal the other group or whomping big mirrors to capture enough light to make it work. Cool! That means potentially different solutions on moon and planet and the need to develop time tables. I'll leave all that to an exercise for the OP.
In the end what gets sent through the portal is a parabolic mirror of reasonable size (obviously limited by the size of the portal), the basic means of creating dots and dashes (possibly just a piece of cloth) and instructions for when to signal and where to point the mirror.
How to point the mirror? Put a hole in the bottom of the dish and some thin struts from the edges to a point above that hole. Not unlike what you see with big radio antennae today, but for a different purpose. Look through the hole, use the convergence of the thin struts as an iron sight, point the dish as best one can to the right spot. At least you won't be signalling South America when you were trying to signal France.
End result: slow but predictable communication
Traditional Morse Code is letters and numbers and it's not as tedious as one might think. Consider the chart, below (click to enlarge), which makes deciphering what the incoming letter might be a lot quicker. Add to that the basic reality that one can predict what the word will be more and more with each passing letter and it gets quicker.
But I can easily imagine a series of complex code meanings: a series of dots and dashes that mean whole paragraphs, if necessary.
However, remember that thought I asked you to hold? Keep in mind that it's really only the moon that needs to use this solution. The planet can always toss very lengthy answers back through the portal. Whole libraries, if necessary. Along with supplies and all kinds of things.
Which is good, because eventually some idiot is going to send a message along the lines of, "Sucks to be you, mate! BTW, your wife just bore you a son! Kinda looks like you! Maybe..."
They can't... not for a long time anyways
As much as I hate to admit it, I really don't see an easy solution here. Yes, there are ways they could communicate eventually. Large scale manmade features as Dutch suggested. Giant mirrors could be another method. The problem is you have only 20 people.
There just isn't enough manpower to construct things big enough to been seen by the people back on the planet. Additionally those twenty explorers will have barely any free time to figure out the communication problem to start with. Their priorities will be finding food and shelter (I'm assuming the moon is habitable). Did they bring crop seeds with them? If not then they're stuck with an even more time consuming hunter gatherer lifestyle. And when one of them inevitably gets sick or injured? Now the others have to pick up the additional workload.
Got food and shelter sorted out? Now they'll need to rebuild an industrial base from scratch. Whatever tools the brought with them will wear out eventually so they'll need a blacksmith and carpenters workshop at the very least. Then they'll need timber, charcoal production, ore mines, and metal smelters just to supply the blacksmiths and carpenters. Even if done on a very small scale this is still going to require a huge amount of work.
But the problems don't stop there. Do the people on the planet even know the magical gates lead to the moon? And if they do then do they know which part of the moon the gates lead to? Even if they know that as well they'll still be stuck trying to spot absolutely miniscule differences on the moon's surface with premodern telescopes.
Maybe in a few generations when the moon colony has enough people and resources it can begin working on constructs big enough to been seen by the planet based population. But I don't think it could be done by the original explorers in any reasonable timeframe given how few they are.
Remodel the surface to show features big enough to be seen from the home planet. Rinse and repeat until the message is sent in its entirety. Sort of Nazca lines on hormones. E.g. first engrave an H, then wipe and engrave an E, then wipe and engrave an L, then wipe and engrave an L and then wipe and engrave an O to say HELLO.
Slow as hell, though.
You could do this with a combination of a large feature to attract attention to a specific point on the surface, and a smaller, but still large array of heliograph mirrors.
Percival Lowell and Ernesto Schiaparelli thought they had detected canals on Mars. Lowell interpreted these as a titanic engineering project to irrigate the dry equator using polar ice. But this was Mars, which is a lot further away than the Moon, and they thought they could.
A good 8" telescope can resolve high-contrast features down to a mile across. Less, if they have a good telescope. Make a 1-mile wide 'L' shape on the moon and place the heliograph on the angle of the 'L'. This feature does not have to be permanent: marking light ground with soot, or ploughing might be enough to get people looking in the right place.
Do the people back on the home planet know they went to the moon? I guess not, or they may have predetermined a place to signal from, supplied the party with mirrors, and have a larger telescope to look for a signal. My 8" telescope is a good amateur size.
Lighthouses
Gigantic solar system sized phased laser array constructs in stable orbits that send one directional messages via laser towards suspected habitable words.
Basically, you have a computer, that calculates the position and climate of far away exo-planets and if they are deemed hatiable (like venus, earth and mars in the solar system), it simulates the future position of the planet and fires a message laser toward this. This message laser is pulsed and has encoded the senders stellar location, some basic encoding scheme and other important stuff to "connect" to the light house interstellar network.
On the target planet, years later, a gigantic glowing disc of light appears, racing at the rotational speed of the planet (Earth 1,600 km/h) over clouds and ground. If the observation is that good, that it can compensate for the rotation, the disc may even appear stationary, although that is unlikely to attract the attention of any hightech civilization that could record and decode the impulses. Better to go back and forth fast to get attention first and then slow down to rotational speed so a sensor can catch up and deploy the message then can be recorded.
Once the target civilization replies, with the communication protocoll (which is TPC/IP) from the Unidentified Message Object, it is integrated into the interstellar civilization. Which deeply distrusts each other, which is why nobody has the lighthouses and routers in his/her home system.
