Discrete v/s Continuum Instruments
Disclaimer: Anything that I write below is from the past 6 years of my self-training in music. I have never had a formal education in music. I have almost zero experience with percussion.
I divide all the non-percussive instruments into 2 broad categories - Discrete and Continuum.
Discrete instruments are those which have clear markings of the natural notes. For example, Guitar is a discrete instrument. It has fret markings at varying distances where the notes would ring out. Flute is discrete too. It has holes that let the air pass and produce the notes. Piano is fully-discrete in the sense that it is impossible to play anything other than discrete notes on it. The easy part of learning to play a discrete instrument is being able to play natural notes. However, the difficult part of it is being able to play bend notes. The 7 notes of a musical scale sound great to human ears, but, they can become monotonic to the listeners. It's the ability to distort those notes through bends that add the feel and emotions to the playing. There are many many forms of bends - a quarter bend will go 1/4 up or down, a semi bend will go 1/2 up or down, a vibrato bends the note up and down rapidly centering around the note, and a slide can temporarily cover all the frequencies between two notes in a rapid flow. Bends are difficult on discrete instruments. For example, on a harmonica, it requires special tongue action to let the air pass in a particular direction that produces a bend note. A piano cannot play bend notes at all unless there is a pitch bender pedal attached to it.
On the other hand, continuum instruments don't have markings of the natural notes. The notes need to be spotted purely by practice. For example, Violin (and other instruments of the string family) is necessarily fretless instrument. It can take an insane amount of practice to be able to place fingers are the exact points where the natural notes would ring out. Continuum fingerboard is another such digital instrument. However, once you are able to find the location of notes, both natural and bend notes are almost going to take the same amount of effort.
From my experience, every musician should learn at least one discrete and one continuum instrument. A discrete instrument would help you quickly land on the natural notes but will require a lot of practice to bend the natural notes to get the feel right. This will help you master the bend notes. On the other hand, a continuum instrument would require putting sheer effort into getting the natural notes right as you would almost always hit higher or lower than the actual notes. This will help you master the natural notes.
I do not have any recommendations for a specific discrete instrument. But I do have one for continuum - Vocal. Vocal is a continuum instrument. From my experience, I can say that vocal is the highest form of continuum-ness that a musician can ask for. Violins have a clamp at the beginning of the fretboard. On a tuned violin, you can always play the open string to figure out at least one note and then your relative pitch can help you find the others. Vocal has no such clamp. In a solo vocal, once you lose track of your root, you are lost completely. There is no way to get back to the root.
I would recommend every musician to learn vocals, no matter what instruments one plays. I saw a huge improvement in my guitar playing after I started learning vocals. Vocals will help you master relative pitch at a much faster pace than ever. It also improves other aspects of musicianship - perfection & feel.