Independent significance of intercept and slope is done using a t-test.

Joint significance of intercept and slope is done using the F-test.

Determining significance basically boils down to check whether the coefficient in question differs from a required (usually 0) value with a pre-set certainty.

You use an F-test to determine whether the slope is not zero (in a simple regression) or slopes are not all zero (in a multiple regression); _ the F-test does not include the intercept _. Indeed, there is no single test for whether the slope(s) and the intercept are not all zero.

You’ll usually calculate a t-statistic for the intercept. You can either calculate the t-critical values using α and compare the t-statistic to the t-critical values, or they’ll give you a p-value and you can compare that to α.