You perform the actions stated on a card in sequence. For some spells and abilities, you'll surveil last. For others, you'll surveil and then perform other actions.

Some spells that instruct you to surveil require targets. You can't cast a spell without choosing legal targets. If all of those targets become illegal, the spell doesn't resolve and you won't surveil.

When you surveil, you may put all the cards you look at back on top of your library, you may put all of those cards into your graveyard, or you may put some of those cards on top and the rest of them into your graveyard.

You'll surveil before resolving any abilities that trigger on the target creature dying.

Deadly Visit can target a creature that's indestructible. It won't be destroyed, but you'll surveil.