If the adventurers are casually regarding a seemingly lifeless gargoyle that then animates to attack them, you should be rolling initiative first before any attacks are made. When dealing with a situation in which a creature could be surprised, it is important to know when to call for initiative.
This is the second big mistake a lot of people make when using the surprise mechanic. It is also important to note that attacking a surprised creature does not necessarily mean you have advantage on the attack roll. Surprise and advantage often go hand in hand thanks to the prevalence of ambushes by hidden attackers, but the two mechanics are not bound to each other. The major differences are that there is no limitation on what kinds of actions can be taken in this round (in Third Edition, you only had one action in the surprise round) and, in keeping with Fifth Edition’s attempts to streamline the game, the entire thing has been wrapped up into the first round of combat. If this seems familiar to some players, it’s because this is very similar to how the surprise round worked back in Third Edition. A member of a group can be surprised even if the other members aren’t. If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter. Otherwise, the DM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. If neither side tries to be stealthy, they automatically notice each other. The DM determines who might be surprised. The following excerpt page 189 explains how it works: Instead, surprise is covered in chapter 9, “Combat”.
In fact, all conditions appear in Appendix A: “Conditions” in the Player’s Handbook, and a brief glance at that section will reveal that surprised isn’t there. Many players erroneously call surprised a condition.
Let’s start with a very common misconception.