First off, if you are renting where you live, then there's a limit on what you can do that will be effective. The most effective things you can do require modifications and therefore ownership.
A little bit of physics: When you play, the piano vibrates the air, the air hits the wall and starts the wall vibrating, some of the energy gets absorbed by the wall and some of it gets passed through the wall and vibrates the air on the other side of it, which then vibrates your neighbor's eardrums and they can hear you playing.
We don't want to change how the piano works, and we can't change anything on your neighbor's side of the wall, so we have to either change the transmission through air from the piano to the wall in your apartment, or we have to change the wall.
The essential property that we need to change in order to stop or diminish acoustic vibrations is mass. We want to put as much mass as possible (ideally mass that is effective at converting acoustic energy into heat) in between the piano and your neighbor. We have narrowed down our options to adding mass to your side of the wall, or adding mass in the air between the piano and the wall in your apartment.
Since sound waves diffuse around things, we can't just add a lump of mass somewhere in the room - it has to be spread out and be as large as possible. So in terms of attacking the air space between the piano and the wall, our best option is another wall. This can be a temporary wall (and this is pretty much your only option if you rent) or a permanent wall.
Temporary walls (usually called gobos, not to be confused with the concert lighting devices of the same name) can be purchased in many different configurations but are often made from materials available in hardware stores. Note that most gobos are designed to absorb and diffuse sound and less designed to stop transmission, with the exception of a specific type of gobo called a drum shield. If you are renting (or if this seems like the easiest thing your situation), a drum shield may be a very good solution to your problem.
If you own and you want to go with the "nuclear option", then you can build one or more walls within the room. The particular design for these walls is another question entirely, but this answer will touch on design elements below. Actually building a room within a room for isolation is the ultimate step. For a piano, that's probably overkill.
If you own and you want to reduce the sound transmitted through the wall itself, then again, you'll want to add mass. Not all mass is created equally, though. It turns out that gypsum board (also called drywall, sheet rock, or wallboard) has pretty good sound absorption for its weight, and of course is pre-made to be a wall construction material (particularly note that gypsum board is also effective at slowing the spread of fire). So, merely adding a layer of gypsum board (5/8" thickness seems to be popular for this) to the inside of the existing wall can make a big difference. One simple and affordable way to make a bigger difference when adding gypsum board is to use Green Glue to attach the gypsum board to the existing wall. Green Glue is specifically designed to both easily add a little more mass and to maximize the sound energy that is converted to heat by the augmented wall. Green Glue with 5/8" gypsum board is a very reasonable and effective option for augmenting the wall between you and your neighbor to reduce sound transmission.
If you do feel like building another wall is the way you want to go, there are likely to be a lot of code implications and other challenges. You could merely build half a wall right in front of the existing wall, with the interior side composed of two layers of gypsum board with Green Glue in between. If done correctly, that could be as or even more effective than building an entire wall, and it would also minimize the impact on the room size.
Yes, sound can go through the ceiling and then sideways and then back down into your neighbor's apartment, but that involves travelling through three walls instead of just one. I wouldn't worry too much about this unless and until you have taken some action to mitigate the more direct transmission and then determined that the "upper" route is also a problem. If it is, you'll notice drum shields are available with "ceilings" and gypsum board and green glue can be applied to ceilings as well. You could also put heavy rugs and or carpet on the floor above.