The U.S. federal judge -- who President Biden nominated as the best person to next serve on the high court -- finally got a chance to speak Monday on Capitol Hill after hearing from senators on both sides of the aisle about what to expect over the next several days.

During her long-awaited opening remarks ... Jackson took a tender moment to acknowledge her family and how much they've meant to her in getting to this potentially historic milestone.

If confirmed, she would become the first Black female justice to serve on the bench -- and with that in mind, the judge gave it up for her hubby of 25 years, Patrick, as well as her daughters, Talia and Leila, whom she thanked in what made for a pretty emotional scene.

It was reflective of when Biden first announced her as his pick ... there, too, she took time to honor her brood.

As for other highlights from Day 1 ... it was a mixed bag of both praise and prefacing of expectations.

Dems gave props to Jackson for making it this far and possibly becoming a trailblazer. Meanwhile, Republicans preemptively brought up what they're anticipating -- namely, being accused of racism for asking tough questions.

They also made sure to repeatedly promise ... they would NOT do to her what they claim Democrats did to Brett Kavanaugh during his confirmation hearing -- which they label a character assassination. This, of course, before they've even asked her one question.