If I'm in synagogue and someone asks me if I can "open the Aron" or "take out the Sefer Torah," what will I be expected to do, and when?
It depends on the synagogue, and depends what you're being asked to do. I'll assume the synagogue is Ashkenazic.
If you're being asked to open the aron (cabinet holding the Torah scrolls) for a particular prayer, but the scrolls are not to be removed, then the first step is to approach the aron at the appropriate time. (Actually, as mentioned in the other answer, some synagogues will want you to wear a talis for the occasion. If you're not wearing one already, then ask the person who gives you the job whether you need one and where you can get one.) The time to approach will be signaled by the gabay (usually the person who asked you), at least if you ask him (when he asks you to do the job) to signal you when you should do it. Or he might tell you ahead of time to do it at a certain point in the service, in which case you'll have to keep track. Once you're near the aron, you touch it and kiss your hand (or kiss it directly), wait until the appropriate point (signaled by having reached the right point of the service, by someone's signaling you (perhaps the rabbi, who usually prays near the aron), or by the entire two-hundred-person congregation's staring at you expectantly), and then open the aron. More on the mechanics of this in a bit. After the prayer is concluded, you close the aron, touch it and kiss your hand (or kiss it directly), and return to your place (not by the shortest route). People may tell you "chazak uvaruch" or "y'yasher kochacha" or "shkoyach" (all good wishes (blessings) told to someone who did a good job) or similar, and an appropriate reply is "baruch tihye" (="you should be blessed [too]") or just "thanks". Return the talis to wherever (or whomever) you got it from, and your job is done.
If you're asked to also remove the Torah scroll from the aron, then, in addition to the above, you, well, you remove the Torah scroll from the aron. You can check with the rabbi (who, again, is usually praying near the aron) about precisely when to remove it: some synagogues like to do so before, and others after, a prayer called B'rich Sh'meh. Also check with him which to remove, if there's more than one (except that if there's only one without a belt visibly around it, then that's the one to remove). When he gives you the nod, you remove it with two hands, kiss it, hold it in your right (even if you're a lefty, unless you are afraid you may drop it, in which case hold it in your stronger hand) or both hands, leaning against your right shoulder, and when the leader of the prayers approaches you, hand it to him with both hands. You hold it, and hand it over, in a vertical position. If it's behind a cordon, you will probably have to detach one end of the cordon first, in which case reattach it after handing the scroll over if there's another scroll behind the same cordon, or after returning that scroll to the aron if there isn't. You close the aron after handing the scroll over also. Then the leader of the prayers will carry the scroll, and you follow him until he puts it down on its table. Get the gabay to tell you when to return the scroll to the aron and where to stand to get it (in most synagogues, near the aron); then reopen the aron, detach the cordon if necessary, receive the scroll, kiss it, place it in, and reattach the cordon. In some synagogues you can close the aron when you're done; others will want to wait until the prayer recited when the scroll is being returned to the aron is concluded; usually (in my experience) the former don't sing that prayer and the latter do. The rest of the instructions are the same as above (in the case that you're not handling a Torah scroll).
On occasion, the rabbi or someone will tell you that there two Torah scrolls being removed from the aron that day. In that case, in some synagogues, you'll hand the second scroll to someone also, and proceed as above. In others, you have to hold it with one hand while you close the aron (and put the cordon back) with the other, which can be difficult. In that case, if you are afraid you will drop the scroll, ask the rabbi or someone else standing there to close the aron for you, explaining that you can't because of the scroll: he'll be glad you asked and will do it. (On Chanuka there may even be three scrolls to take out, in which case someone else will be there to take the second and either you or another will hold the third, as above.)
The mechanics of opening (and closing) the aron: Usually there's a curtain in front of the aron; this must be moved aside. If there's a drawstring hanging down in front of the aron on one side, then it probably controls the curtain; otherwise, just shove the curtain aside manually (it'll be on a curtain rod). Then there'll be a door. Sometimes it's a sliding door, sometimes a hinged door. Sometimes it's a safe. You'll be able to figure out how to open it, or someone there will tell you. Occasionally there's a curtain inside the doors instead of outside, in which case open it as described above for an exterior curtain.
I have not included a discussion of Simchas Tora, a day when many scrolls are removed and the procedure is somewhat different, nor of Hosha'na Raba, a day when, in many synagogues, several scrolls are removed, but also with a different procedure.
This answer is without citations to sources, but is based on what I've seen people do. Not everyone (and not every synagogue) does everything I've mentioned; not necessarily does everything I've mentioned have a basis in halacha. For practical halachic guidance, as always, CYLOR; let this be only a starting point of your discussion with him.