Learn German: Ich bin ziemlich müde.

You can place it right after the verb: Ich bin selbst ziemlich müde. This emphasizes that you, too, are tired, perhaps in response to someone else complaining about being tired. You could also use selber, which means the exact same thing but is more common in everyday speech.

Does ziemlich mean 'very' or 'a little bit' here?

It falls right in the middle! Ziemlich translates best to 'rather', 'quite', or 'fairly'. It means you are definitely tired, but not completely exhausted (sehr müde).

If I started the sentence with ziemlich müde, where would the verb go?

The verb must stay in the second grammatical position. If you want to emphasize your tiredness by moving that phrase to the front, you would say: Ziemlich müde bin ich. This sounds highly emphatic, but it perfectly follows the strict V2 (verb-second) rule!

Is müde related to any other common vocabulary?

Yes! You might know the noun die Müdigkeit, which means 'tiredness' or 'fatigue'. Adding -keit or -heit to an adjective is a very common way to turn it into a feminine noun in German.