Playing UNO FLIP!, we flipped the cards (the cards on hand, Draw Pile and Discard Pile) and now a yellow Skip Card appeared on top of the Discard Pile.
Should the next person's turn be skipped or not?
No, you don't take any actions based on the card you see after a flip.
Here's the full rules of what happens when you play a flip card:
Flip Card - when you play this card, everything flips over from the Light Side to the Dark Side. Once the Flip card has been played, flip over the Discard Pile (the card just played will now be on the bottom), then the Draw Pile, then everyone’s hands must flip to the other side. This new side will remain in play until another Flip card is played, changing it again. This card may only be played on a matching color or on another Flip card.
No mention of skipping or reversing or taking any other actions based on what card is on top after the flip.
The face-up card on the discard pile simply determines the next legal play:
Whether playing the Light Side or the Dark Side, on your turn, you must match a card from your hand to the card on the top of the DISCARD pile, either by number, color or symbol (symbols represent Action Cards; see FUNCTIONS OF ACTION CARDS).
So the next player either plays a yellow or skip card (matching the top card), or a wild.
The UNO Flip! rules (from 2020) now clearly state what to do.
When the deck is flipped: