tamburo

One root, tambur-, and no -ur- in it, because Esperanto has no -ur- suffix. That is the whole point of listing it: a learner meeting -ulo, -ilo, -aro and -ejo in quick succession starts to see suffixes in every final syllable, and tamb-ur-o is the kind of parse that follows. There is no tambo. Check backwards before you split, the same test that disposes of karoto and perspektivo.

Words that actually derive from it. Tamburi is to drum, and it is a proper verb: you can tamburi per la fingroj sur la tablo, drum your fingers on the table. A tamburisto is a drummer. Tamburbastoneto is a drumstick. Tamburado is drumming as an activity or as a noise. All of those are built with real suffixes on a real root, which is the contrast worth noticing.

Vocabulary around the instrument. The drum head or skin is the membrano, the whole kit is a drumaro or tamburaro, and the big orchestral kettledrums are timbaloj. Ritmo at rank 2226 is the rhythm; the beat you count is the takto. If you want the sound rather than the object, Esperanto has an onomatopoeic stock for it, and interjections and onomatopoeia covers how the language handles noises generally.

Historically the drum was a signal before it was music. A tambura signalo carried orders across a battlefield further and more reliably than a shouting officer, which is why the instrument travelled with armies and why so many European languages ended up borrowing the same eastern word for it.