Learn Turkish: Film biraz sıkıcı. - The film is a little boring.

Turkish doesn't have a direct equivalent for the word 'the'. When a noun like film is the subject of the sentence, it is usually understood as specific (the film) from the context. If you wanted to say 'a film', you would add the word bir (one/a) before it.

Where exactly does biraz go in a sentence?

Just like 'a little' or 'a bit' in English, biraz comes directly before the adjective it describes. Here, it sits right before sıkıcı (boring) to tell us the degree of boredom.

I don't see a word for 'is'. How does the sentence mean 'The film IS a little boring'?

In Turkish, 'to be' (am/is/are) is usually added as a personal suffix to the end of the last word. However, for the third person ('he', 'she', or 'it' like our film here), there is often no visible suffix at all! Simply stating the words together is enough to mean 'it is'.

Can I use sıkıcı to say that I am bored?

No, sıkıcı only means 'boring' (describing the thing that causes the boredom). Just like in English, 'bored' and 'boring' are different words in Turkish. To say you are bored, you would use a completely different word, like the verb sıkıldım (I got bored).