"The Gifts of God, for the people of God."
That's your cue.
You stand up and deliberately walk to your place along with the other chalice bearers and acolytes in front of the altar. Communion is given to you first, and then you are handed a chalice and a small cloth. As this is taking place, as you're hearing the words "The Body of Christ, the bread of Heaven" you hear the faithful gathering on their knees at the altar rail behind you.
This is when it gets real.
Assuming you're the only chalice bearer for a small service, it's relatively simple. You wait for the priest to get about three or four people ahead (quietly intoning) and then immediately start in behind him, offering the wine with your own small speech.
The priest gets the easy part. Giving someone a wafer or a piece of bread from the Fraction is a function of pressing it either into the palm of the hand or onto the tongue if the parishioner is more Anglo-Catholic in bent. On the other hand, you have to approach the person, determine what they wish to do, and then follow along.
The easiest ones to deal with are those who intinct. (Intinction is the deliberate co-mingling of the host and the chalice wine by slightly dipping the host into the offered chalice, taking away the wine "soaked" slightly in the wafer). You simply drop the cup down enough for the parishioner to dip their host in the chalice without doing too vigorous a wrist bend, say something solemn ("the blood of Christ, the cup of salvation") and then move on. Unless said parishioner drops the host into the chalice entirely, in which case you quickly but without hurry summon a nearby acolyte to literally fish it out with a special spoon.
Those who wish to drink from the cup, are most welcome to do so. Hopefully they're helpful and as you reach towards them (hopefully they're tall enough you don't have to stoop too far) take the cup in hand and guide the angle of the chalice to take a small amount, nodding slightly and pushing back when they've drunk their portion. When they don't, especially those from certain cultures who will not touch the chalice at all and will actually put their hands behind their backs, you have to hope you're not about to inadvertently pour very staining material down the front of the best clothes they own, or worse, that you don't tip the chalice sufficiently, and they're cheated of that part of the Eucharist. So you're sweating slightly, especially if they're very short, leaning and trying to gauge the liquid's progress inside the chalice, hoping you're timing it right so as neither to cheat them with a touch of the wine to the lips or worse, filling their mouth to excess. But you can't show it. You have to be dignified, and say your bit, nod and move on.
Some people choose to only accept the bread portion of the Eucharist, because of certain circumstances (food allergy, concerns about alcohol consumption/alcoholism, personal preference). Many will signal you to move on by crossing their arms over their chest or a small wave of the head. Others will leave you there to guess they wish to be skipped over, which can be awkward. (It's important to note that one can (in the Anglican tradition) consider themselves in communion absent the consumption of either part. Though some churches accomodate parishioners with a gluten allergy with specially prepared communion hosts, some choose not to either eat or drink or either, and from a religious standpoint, that's OK.)
Once this has taken place you take the cloth in your hand and, using a part that's not yet touched anything, wipe the rim of the chalice that the lips have touched, turn the chalice slightly to present a new part of it to the next parishioner, and begin again. The problem is that the cloth is a tightly folded and stiff material, and you want to do it right - namely using a new part of the exterior edge every time, without either contaminating one part with another part or getting any of it on your robes. Also, you're "pinching" it beween your thumb and fingers and hoping it doesn't fall, dip into the wine, or what have you. And you're trying to keep whatever is going to stain the cloth to the outside. If you stain the embossed embroidery in the middle, which is difficult to see in the ambient light, you're going to make the Altar Guild quite unhappy, because that embroidery is hard to get totally clean.
At the same time you have a few seconds to gauge how much wine you have left, because you don't want to present anyone with a nearly empty chalice.
And in the meantime, the priest, having performed his much quicker part of the ceremony, is streaking ahead and in danger of coming back around behind you and having to stand there waiting for you to get ahead, without looking like he's impatiently standing there waiting for you to get ahead, all the while that there's a literal line of people waiting to come up to the altar.
Want to make it more fun? Multiple chalice bearers and multiple priests at a larger service. That involves a complicated figure-8 move in which you and your neighbour perform your duties, and then move two parishioners ahead with him or her right beside you. If you cannot do this roughly at the same time he or she (or you if you're following someone else) have to figure out where you are in space and time, especially if his or her next move is to circle back and start again from the farthest part of his or her section of the altar rail. All the while there are multiple people moving in the confined space between the altar and the altar rail, and given this is a pretty important part of the service, you don't want any kind of collisions, confusions or inadvertent Keystone Kops routines.
It's like being a cross between a honey bee and a duck - the bee doing a complicated communicative dance in the middle of the hive while swarmed by others, and like a duck in that you're presenting a gliding, calm exterior with ritual reverence while doing a lot of things under the surface - being aware of your surroundings, the person in front of you, and the practical matters at hand. It requires a bit of practice when it's simple, and when things go odd, like a parishioner collapsing over the altar rail or someone dropping something, you have to juggle it in real time.
The best version of this is a large enough service that you have yet more servers at the base of the altar. Most people go to the altar, so you have a limited flow of people, can stand there inviting people to come get Communion, and the vast majority of people intinct. You're also usually the ones to be tapped to go out into the congregation and offer Communion to those too old, or infirm or otherwise unable to come up to the altar in any way, which is a wonderful thing to do in itself.
It takes practice, it takes discernment, and it takes someone who can juggle a few things at once. But it's what you trained for, and it's just what you do.