I have some a cappella songs that I sing frequently with friends. I would like to write down on sheet music a piano or saxophone accompaniment, what would be the steps that you would take to do such.
What you are doing is called music transcription. You will take a song and transcribe the music onto paper. This is sometimes called arranging, and it requires a good ear, practice, and patience. It is not easy but millions of people do it.
1. Figure out the chords and melody
This is best done on a piano or guitar and requires some basic music theory knowledge of chords. The melody can be done however without knowledge of chords, but it will be difficult. There are online tools that can help you with these things. Some words to google may be: chord detector, music transcription software, etc. I don't use these tools so I am unfamiliar, but they do exist. If you do have sufficient music theory knowledge you can transcribe with your ear and by guessing and checking. Major and minor chords are very distinct, and melodies are comprised entirely of intervals that also have very distinct sounds. You may want to look on YouTube for tutorials on how to transcribe music. For many popular songs there are already chord charts (lead sheet without melody) and lead sheets (chord charts with melody) online that people already transcribed and have shared. This is probably the easiest solution to your problem.
2. Write the music down
Hopefully, you know which notes on your instrument go with what notes on sheet music. However, it's not too hard to learn. For piano/saxophone I would suggest a lead sheet. This would allow a good saxophone or good piano player to play music. Keep in mind however the piano player would need to understand chords and the music would probably need to be transposed for the saxophone player. Transposing is not the only challenge for arranging for a saxophone. The single saxophone can often take the stage, so be careful not to overwhelm the vocals with a saxophone.
3. My piano player doesn't know how to read chords
Well, your piano player should learn, but you can write chords onto sheet music. This requires you to write all the chords in score vertically from each other.
4. I have terrible handwriting
You can write sheet music electronically. It is a bit more time consuming but it comes out neater.
- lilypond.org/ (I use this one but it has a really steep learning curve)
Google and YouTube are your friends in this endeavor, but remember it takes time practice and skill to do these things.