Learn Russian: Верно, я тоже так думаю.

In this sentence, так means 'so' in the sense of 'this way' or 'like this'. It is not setting up a result clause (like 'I think so much that my head hurts'), which is when you would need что. It simply stands alone as an adverb describing how you think.

Why do we use так to mean 'so' in я тоже так думаю?

In English, 'I think so' implies 'I think that this is the case'. In Russian, you literally say 'I also think in this way' or 'thus'. Так is an adverb of manner here, pointing back to whatever was just said.

Could I start the sentence with Правильно instead of Верно?

Absolutely. When agreeing with someone, верно (true/right) and правильно (correct) are often used interchangeably. Верно has a slight nuance of 'That is true', while правильно sounds more like 'That is correct'.

Does тоже have to go before так?

Russian word order is flexible, but я тоже is a very natural conversational block meaning 'me too' or 'I also'. Putting так right before the verb (так думаю) emphasizes the action of thinking 'this way'. You might hear я так тоже думаю, but the order shown here is extremely common and natural.