Now that you have defeated Falkner, there is a little bit of a side trip that you can take before continuing on the main journey. But first, after you leave the gym, you'll get a call from Professor Elm, who wants you to carry the Mystery Egg that he received from Mr. Pokémon. Elm has sent one of his assistants to the Violet City pokémart to give you the egg, so go there and get the egg from him.
When you go outside, a mysterious Kimono Lady will talk to you about the egg and about Professor Elm, before wandering off. After you walk around for a while, the egg will hatch into a Togepi! Once Togepi hatches, you can go to Professor Elm back in New Bark Town, and you will get an Everstone from him. If you give the Everstone to a pokémon, that pokémon will not evolve, even if it reaches a level where it normally would, or if its happiness is enough to evolve on the next level.
Find the west exit of town and go out. Talk to the people here, and one will give you HM06 Rock Smash. This is a very useful move, and now that you have the Zephyr Badge, you can use it outside of battle. It can be used to smash rocks, which sometimes block your path. Also, smashing a rock will sometimes start a battle with a wild pokémon that was hiding inside. There are even pokémon that can only be found with Rock Smash. Some rocks even hide items inside, so get smashin'!
If you go here on a Thursday, then Arthur of Thursday will be walking around this area, and will give you a Hard Stone.
Go south from here to discover the mysterious Ruins of Alph. There are some rocks here that you can smash, and you can find some pretty good items here.
Try going into the stone buildings, and if you don't go anywhere, try moving left or right to see if there is a side door that you can go out of. Eventually you will reach a room where a woman tells you about the puzzle on the wall. Go check on it to try it out. You need to rearrange the tiles to make the shape of a pokémon. You can drag the tiles with the stylus, and you can rotate them by double-tapping on them. It shouldn't be too hard to figure out how to make a picture with the tiles.
Once you solve the puzzle, a trap door will open below you and you'll fall to the lower level. There is a scientist guy down here who gives you the Unown Report. What's an Unown? Well, walk around this area for a bit and you will encounter them. Unowns are oddly-shaped pokémon who only know Hidden Power. In fact, although unown is only considered one type of pokémon, they come in many different shapes. And as for Hidden Power, the move type of Hidden Power depends on some secret attributes that are unique to each pokémon, so you can only find out the type of move that it is if you try it out on different types of pokémon and see whether it is super-effective, or not very effective, or what.
Try catching some and then check the unown report. Notice something? Each unown shape seems to represent a letter of the alphabet! Ruins of Alph, alphabet... get it? You might feel inclined to complete your collection of unown by catching all of the different letters, but unfortunately, the unown that are floating around here will only come from part of the alphabet. There are some tasks that you will have to come back and do later on before you can get all of the unowns.
After you're done in here, climb the ladder and then go back to the room where you solved the puzzle. Look at the back wall and you will see some engraved unowns that spell the word ESCAPE. This is a clue to an item that you should use while standing at this spot. I'm sure you can guess which item to use….
An escape rope! It won't take you out of the room like it normally would; instead, a door opens up after you use it. Go through, and you will find some items here. Fall down the hole here to read a secret message spelled out with engraved unown shapes. Finally, drop down another hole and you'll be back in the area where the guy gave you the Unown Report.