There are tons of ways to say this, and which one is preferable depends highly on the context and what you're trying to emphasize.
In this particular context, a common way to express it is simply with で. Like「彼女は画家で詩人です」. You often hear it in the middle of a sentence, like「画家で詩人のAさんは(...)」(examples) .
Another one you could use is 兼{けん}.「彼女は画家兼詩人です」would simply translate to "She is an artist and a poet". This expression is commonly used when a person has been given two different jobs or other forms of 'duty' at the same time.
You can also use「XでもYでもある」as in istrasci's answer. This is often used in the sense of "Not only is she an artist, but she is a poet as well", i.e. to add some emphasis. A version where this aspect is less prominent would be「彼女は画家であり、詩人でもあります」, but here, you would usually put their main occupation (or hobby etc) first. This would correspond to something like "She is an artist, and she is also a poet.", with a formal tone.
A fun way to say this is「Xであると同時にYでもある」(random examples) or in this particular case「彼女は画家であると同時に詩人でもある」. This translates to "She is simultaneously both a painter and a poet". Here, as you would expect, you would be emphasizing the fact that she is both at the same time (although whether this is actually relevant or not would depend on the context). Another similar (more normal) expression would be「XでありながらYでもある」.
If you refer to their line of work as 'what they do' instead of 'what they are', you could also use も. Like「彼女は画家も詩人もされている方です」. This sounds similar to "She (is a person who) both paints and does poetry." You could also use と, like in「彼女は画家と詩人をやっている」, which would more or less correspond to "She paints and does poetry.".