The target will carry out the suggested action as soon as the game's action economy allows
Nothing in the text of suggestion says that it gives the target extra actions or the ability to take actions on your turn. The spell just says that
[The target] pursues the course of action you described to the best of its ability.
So, if what you suggest requires an action to carry out, then the affected target will attempt to take that action on their next turn. Likewise if it requires a bonus action. If your suggestion requires a reaction (e.g. "I suggest you use your shield spell"), they will take that reaction as soon as they are able, which may not be until after their next turn if they have already used their reaction for this turn. If your suggestion is to move somewhere, they will start moving there on their next turn.
Observers only know about what they can observe
As to what knowledge other characters have about the spell, the guiding principle is that characters only know about what they can observe, unless the spell says otherwise (which suggestion does not). They will certainly know that some spell was cast, as long as they are able to hear you, since the spell has a verbal component, which means you must chant the mystic words to cast the spell before stating your suggestion. Given this context, it will be apparent to most intelligent creatures that you have just cast some sort of spell compelling the target to carry out your suggestion. However, note that only your target needs to understand your suggestion, so you might be able to obfuscate your actions by speaking your suggestion in a language that only the target can understand.
Similarly, other characters can only tell whether or not the target failed their save based on what they can observe. Unless they have something like detect magic that can perceive the spell directly, they will have to judge whether the target is affected by the spell based on the character's actions. However, at this point, we get into DM discretion territory. For example, if you suggest that the target move somewhere and they fail the save, the DM would be within their rights to say "the target turns toward the spot you specified and looks like they're about to walk that way", which indicates that the spell has an immediately visible effect on the target's behavior. Or the DM could wait until the target's turn to say what the target does, giving no indication before then. Either way is a valid way to run the game. Similarly, if the target makes their save, they are free to say so to their allies, but the ability to talk on other characters' turns is something that some DMs allow while others don't, and this will affect how soon the target is able to warn their allies.
In addition, there is potentially another way for a character to determine whether the spell took effect: by observing the target's reaction after casting the spell. Your DM could, for example, call for a deception check from you contested by the observer's insight check. If the observer wins the contest, they would infer from your reaction whether or not you believe the spell worked (which you know because you know whether you are maintaining concentration on the spell). Beyond that, your actions can indicate whether you believe the spell worked. For example, if you suggest that the target should hand you an object but then walk away from the target instead of toward them, an observer might see this and conclude that the spell failed.
It is possible to end the spell before it has any effect
Ending the spell, whether via dispel magic, breaking your concentration, or any other means, immediately frees the target from the spell's influence. If this happens before the target's first opportunity to carry out the suggested action, then the spell ends without ever having any effect on the target. Putting this together with all of the above, it is entirely possible that an ally of the target could infer that you have cast suggestion on the target and then take some action to free them from the spell before they can act on that suggestion. They could also take other steps, like grappling the target to stop them from moving in the suggested direction.
Handing over items
Lastly, using the above guidelines, we can address your specific examples:
- If you suggest that they hand you their shield, they will need an action to do so, since a shield requires an action to remove, unless they are simply holding the shield in their hands rather than wielding it. This action is required because a shield is typically fastened to the wielder's arm in some way rather than simply held up in front of them.
- If the character is holding any object in their hand (such as an arcane focus), then they can (probably, subject to DM ruling) let go of it at any time with no action required, allowing you to take the object out of their hands with your free object interaction on the same turn you cast suggestion on them.
Consider readying the spell and releasing it on the target's turn
If you don't want to signal your intent early, you might consider readying the spell and only releasing it on the target's turn. This won't save you from the risk of losing concentration in the meantime, since a readied spell requires concentration. But it does mean that on your turn, you only need to say the mystic words to cast the spell, so no one will know what your suggestion is until you release the spell, and they might not even know what kind of spell you've cast until you release the spell and speak your suggestion on the target's turn.
However, if you decide to use this trick, be very careful with what trigger condition you specify, since the trigger must be perceivable (i.e. "the target starts their turn" is not a valid trigger), and you only release the spell after the trigger occurs. If you aren't careful with your trigger, the target will get to use their action or movement before you can release the spell.