As a Graduate Student in the US (especially international students), ensure (if possible) to tick the following before you graduate from Grad. School:

1. Research - with publication evidence.

2. Teaching - be involved in 1 or 2 courses even if your advisor can fund...

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...you on RA.

3. Mentorship - put yourself out for intentional mentorship of undergrads, especially if your lab attracts undergrads.

4. Leadership - be involved in associations/clubs at different levels. Volunteer at your discipline-based Society.

5. Community Service -

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... get yourself involved in your immediate community - the University, City or the State.

The list is endless. But I think ticking the above would greatly enrich your CV for different career trajectories.

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PS: Some of these things I shared here can also be applied by graduate students in Canada. In UK and Europe, only a few places give that privilege of adding teaching to one's suite of skills as a graduate student.

New addition:

6. Graft writing: A lot of Graduate students do not tend to strive to apply to research funding especially if their labs guarantee them funding for the duration of their programs. I would advise you are intentional about applying to research funding irrespective...

... of your funding status. Apply to departmental fund (conference and research support), University-based fund, discipline-specific research support etc.