I'm pretty sure Obama had a lot more threats than Trump may have had
Actually, no. According to the Secret Service* director, the average number of six to eight threats/day has not changed with the transition of Obama to Trump.
Having observed their agents on protective duty from time to time, I'm also quite certain every threat or even potential threat is investigated.
—
*For those of you outside the United States, the Secret Service is a law-enforcement agency of the U.S. Department of Homeland Security (historically, it was part of the Treasury Department) that is responsible for prosecuting financial crimes such as counterfeiting of currency, as well as providing protection services for certain American and visiting foreign officials.