[noun] / [verb (plain)] + より(も)、むしろ + [sentence] [sentence] + というより、むしろ + [sentence] [noun] + ではなく、むしろ + [noun] [sentence]。むしろ + [sentence]。
むしろ means "rather", "if anything": of two ways to describe something, the second fits better – often the opposite of what the listener expects. The first comes before it with より, というより or ではなく, and むしろ introduces the second.
With より and ほうがいい or たい, むしろ picks the choice the speaker prefers. Both sides of より are actions, so a verb goes in front of it.
At the start of a sentence, むしろ follows a "no" and goes further the other way.
English "rather" also means "fairly", as in "rather tired". むしろ never means that: it always weighs one thing against another. For "fairly", use かなり or けっこう.