This post is not up to date and I will update it soon.
Below you can see my little setup consisting of a Roland RD-800 keyboard, an Avid Mbox audio-interface and a powerful computer. This is basically what you need to start making some piano recordings - and you don't have to throw away all your money on expensive stuff like I have been stupid enough to do ;)
Audio - The Piano
When producing the audio there are generally two approaches. The audio can be produced directly on the computer using MIDI along with a software piano (VST) or the actual sound of the keyboard can be recorded. I always produce using MIDI and this is the method I highly recommend. In order to do that I connect my keyboard to my computer using an audio-interface. In Logic Pro X I record the MIDI file and then I produce the sound with software instruments.
If you have no idea what I'm talking about and what MIDI is then I suggest you read this article - "What is MIDI?". Further, you can read this note about software instruments.
I use the software pianos from Synthogy called IVORY II. More specifically I use the American Concert D (Steinway), Italian Grand (Fazioli) and the three other grand pianos in Synthogy's grand piano package.
The pianos mentioned above sounds beautifully alone, however, I add external reverb to get an even richer sound. Here I use the PCM Reverb bundle from Lexicon which is great. This really affects the sound more than you would believe.
I also use equalizer and other processing tools in order to improve the overall sound quality. I do not have any other experience in the production of music than my works on YT but I feel that I have already learned a lot and keeps getting better.
Audio - Backing tracks
I often use backing tracks for my pieces (primarily strings). I have a lot of software instruments but I primarily use a VST called Cinematic Strings and sometimes also the Emotional Cello from Harmonic Subtones. After the recording of the piano track I produce the strings track and add it to the final mix.
The video production is simple. I have a Canon EOS 550D camera (hoping to upgrade soon) with a 50-200mm Canon lens. I place my keyboard in the middle of my room and add some light sources - voila! While recording the video I record the MIDI file in Logic. Then I edit the audio and add it to the video file afterwards. I edit the video in Final Cut Pro.