Learn Russian: Мой шарф длинный и синий.

Adjective endings in Russian depend on whether the base word ends in a hard or soft sound. The word длинный has a standard hard stem, so it gets the regular masculine ending -ый. The word синий has a soft stem, so it requires the soft masculine ending -ий.

Where is the word for 'is' in this sentence?

In Russian, the present tense of the verb 'to be' is almost always left out. So instead of literally saying 'My scarf is long', you just express it as 'My scarf long'.

How do the other words in this sentence prove that шарф is a masculine noun?

You can tell шарф is masculine because all the words connected to it match that gender. The possessive pronoun Мой and the adjectives длинный and синий are all specifically in their masculine singular forms.

Does the meaning change if I put the adjectives before the noun, like Мой длинный и синий шарф?

Yes, changing the word order turns this complete sentence into a simple descriptive phrase. That order means 'My long and blue scarf', rather than stating the fact that 'My scarf is long and blue'. In Russian, placing the adjectives after the noun helps do the job of the missing verb 'is'.