Read Genesis 2:1-3 (Vayechulu) on Sefaria for the surrounding biblical/liturgical context.
The Vayechulu paragraph is unique: it is one of the very few biblical passages that the rabbis instituted as a witness statement, meaning that reciting it on Friday night fulfills a halachic role parallel to giving courtroom testimony. The two witnesses to creation are the human community (saying Vayechulu) and the natural Shabbat itself (the day's distinct quality from the other six). Saying these three verses out loud is, technically, sanctifying the entire week's claim to a Creator.
The third part, the Mekadesh haShabbat blessing, has six theological moves in ~50 words:
1. Sanctification: "who sanctified us with His commandments", the standard asher kid'shanu opening.
2. Election: "and was pleased with us", God chose Israel.
3. Inheritance: "and gave us His holy Sabbath as an inheritance", Shabbat is not earned, it's bequeathed.
4. Remembrance of creation: "a memorial of the work of creation", Shabbat as the weekly creation anniversary.
5. Remembrance of the Exodus: "a memorial of the going out from Egypt", Shabbat as a freedom-from-slavery declaration.
6. Closing: "Blessed are You, LORD, who sanctifies the Sabbath".
Creation + Exodus + present sanctification all in one paragraph. The Kiddush is theologically dense by design, every Friday night, every Jew is briefed on the cosmic premises of the next 25 hours.
The cup should be at least a revi'it (~86ml). Many homes use a dedicated silver or crystal Kiddush cup. The wine should fill it to overflowing slightly, the overflow is symbolic of overflowing blessing. Hold the cup in the right palm with four fingers underneath supporting and the thumb on the side.
The person who says Kiddush drinks at least the majority of the cup (a melo lugmav, "cheekful"); others take a small sip. Pouring extras from the Kiddush cup for everyone present is customary, especially for the children.
If wine is unavailable, Kiddush can be said over bread (challah) instead, in which case the wine blessing is replaced by HaMotzi, and the sequence shifts. But the standard form is over wine.