Press your lips together to stop the air, then release. No vocal cord vibration.

Americans pronounce part as PART (/pɑrt/). You'll hear it in sentences like "The star is far from the hard part" or "I understand the last part of your plan" — more examples below.
Record yourself saying "part" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Press your lips together to stop the air, then release. No vocal cord vibration.

Open wide for the 'ah' vowel. Lift the tongue back and up while flaring the lips for the 'r'.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "part", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.