how much it would cost to take a taxi from taipei to taichung and how long is the trip?

It is around 170km between the two cities, so it is generaly hard to catch a taxi for the trip. Though you wil luckily find one, it will charge very high. Therefore, I suggest you take high-speed train to travel. It costs 765NTD and 1,095NTD respectively for the standard seat and business class seat per person.