Learn Russian: Экзамен начинается утром.

Yes! Using the compound future tense (будет + the infinitive начинаться) is perfectly fine here to say 'will be beginning'. However, just like in English ('The exam begins in the morning'), Russian frequently uses the present tense for scheduled future events.

Why does the verb начинается end in -ся?

The -ся ending makes the verb reflexive. In Russian, the exam doesn't start something else; it 'starts itself.' If you used начинает without the -ся, it would mean the exam is starting another object, which doesn't make sense!

Why is the word for morning spelled утром instead of утро?

Утро is the basic noun for 'morning'. Changing the ending to -ом puts the word into the Instrumental case. When used with times of day, this turns the noun into a 'when' word. So, утром translates to 'in the morning'.

How do I pronounce the -тся ending in начинается?

Even though it is spelled with -т- and -с-, the ending -тся is always pronounced like -ца (sounding like 'tsa'). The whole word sounds like 'nachi-NA-ye-tsa'.