The following answer is based on my experience in Orthodox, Ashkenazic congregations, primarily in the United states, and incorporates elements from other answerers. I am marking this answer as a "community wiki," which means that anyone with 100 reputation points can edit it, so people with experience in other communities can provide their perspective.
To know ahead of time
- Are you a Kohen, a Levi, or a Yisrael? This will determine which Aliyot you're eligible for. It's dependent on your father's status. If he doesn't know it, consult your Rabbi.
- Know your Hebrew name and your father's Hebrew name. This is used for calling you to the Torah, and, on Shabbat, for blessing you afterward.
- Know the Hebrew names of your closest relatives and their fathers' Hebrew names. On Shabbat and holidays, in some congregations, you will have a chance to bless other people after your aliya.
- Know how to read the blessings before and after the reading. Most synagogues will have a sheet on the bima with the blessings in Hebrew and probably also transliterated English. You don't need any particular tune, but if you're not familiar with the words yet, it's probably worth practicing them ahead of time. If you get up there and find there's no sheet with the blessings, the blessings are also found in the siddur; you can bring a siddur with you to the bima, or if you ask the assistant Gabbai or just indicate your confusion, he should provide you with one. Often a synagogue has a blessing sheet, but they forget to put it in place.
In general, follow all instructions or prompts from the Gabbaim and Torah reader, including if they contradict instructions listed here. All of this happening in the middle of prayer service, so some of their prompts may be in an undertone, or non-verbal. Keep your eyes and ears open, and you'll be okay.
- The Gabbai will call you to the Torah using your Hebrew name and father's name. When you hear this, head toward the bima. If the Gabbai doesn't already know your Hebrew name, he will beckon you to approach first, so that you can provide it.
- If the Gabbai doesn't know your Hebrew name, he may say "Ya'amod ..." ("Arise ...") and then look at you with a questioning expression. Tell him your Hebrew name, "ben" ("son of"), and your father's Hebrew name, loud and clear enough for the Gabbai to hopefully get it on the first time.
- Do you have a Talit on already? If not, ask someone for one, and put it on. Most synagogues have a supply of congregation-owned ones available.
- If you were given a card indicating your Aliya, have it in-hand. If it has folding flaps or paperclips or the like to indicate your donation, make sure it's ready.
- Walk to the bima via the most direct route. All else being equal, choose right turns over left. Be careful not to run into the person leaving the bima -- usually he'll cede right-of-way to you.
- If you have an Aliya card, immediately hand it to either the Gabbai (who will usually be at your left) or the assistant Gabbai (at your right). You can usually see who's waiting to accept it.
- Wait for someone in charge (either a Gabbai or the Torah reader) to uncover the Torah scroll and open it. The Torah reader will indicate where the reading will start. Hold a bit of your Talit over that spot or touch it to the margin nearby, without contacting the print, then kiss the Talit.
- The scroll has two wooden poles, known as "etz chaim"s (or plural, "atzei chaim"). Grasp the right handle with your right hand.
- Turn your head towards the printed blessings while reciting them. This is done so no one gets the mistaken impression that the blessings are contained within the scroll. If you're directed to close the scroll and/or hold both handles, do that instead. Regardless, please don't shove the reader out of the way. Even if the scroll was closed, the reader may pull it open as you're finishing the blessings, to give himself a moment to find his place. Please don't take it personally.
- Recite the beginning blessings. Make sure to give the congregation a chance to respond after the first line before you start the second.
- Theoretically, as the person called to the Torah, it would have been your responsibility to actually read the text with correct vowelization and intonation ... but don't worry, the Torah reader is there to do that for you. However, you do have to pay attention to the reading. It's good to also mouth the words in time with the reading, if you can. Doing this inaudibly will minimize irritation to the reader. Similarly, occasionally a reader may make a mistake or pause to jog his memory. For the reader's sanity, please leave all prompting/correcting/place-pointing to the Gabbai, assistant, or rabbi.
- When the reading is over, do the same point-and-kiss move with a bit of Talit, this time for the end of the reading.
- Recite the post-Aliya blessing.
- Move to the right -- next to the assistant Gabbai. You will remain there until the next person finishes his Aliya.
- On Shabbat and holidays, in most congregations, the Gabbai will recite a "Misheberach" for you, asking God to respond to your Aliya with blessings. If he doesn't remember your Hebrew name, he'll look to you for it again at the appropriate spot in the Misheberach.
- In many congregations, the Misheberach is optionally followed by a second Misheberach for blessing your family. If you opt to have one, you're generally expected to make a donation to the synagogue. If the option is available, the Gabbai will look at you expectantly after the first Misheberach. You should indicate whether you want the second one by nodding or shaking your head.
- In the second Misheberach, the Gabbai will pause and look at you for the Hebrew names of your family members (along with their fathers' names). People typically start with their wife and then go on to children and any other family [or friends] that they want to have acknowledged. Please bear in mind as you decide how many people to name that the congregation is waiting idly during this process, unless you notice that the Torah reader needs a break! You can account for entire branches of a family tree by saying "and his/her whole family." It's nice to acknowledge family members who are present in the synagogue, your host, if you're a guest, and the synagogue crew. If you want something quick, the standard line is "all my family and all those praying here" ("kol mishpachto v'kol mitpal'lim kan.")
- Later in the second Misheberach, the Gabbai will pause and look at you again so that you can name your donation. Some people mention a dollar amount, but the easiest and most discrete option is to simply say "Matanah" (gift). You can decide later how much to give. The Hebrew word for "live" ("chai") has the numeric value 18, so $18 is a common (small) donation; due to inflation, $36 ("double chai") is slowly becoming the new $18. You're likely to see other multiples of 18 ($54, $180, etc.) as popular donations; this is a nice practice, but you can give any amount you like.
- There is also an option in some places to make a special Misheberach for sick relatives. Ask the Gabbai if you can make a Misheberach for a sick person (though you may be asked to make an extra donation for this). In this case, you need the sick person's Hebrew name "bas" the sick person's mother's Hebrew name. If the sick person does not have a severe illness, or the sickness is a continual one (E.G. the person is regularly in the hospital and you have already made regular Misheberach for the sick person), it might be possible to do this instead after all people have been called up. Ask the Gabbai if you're not sure.
After the Aliya
- During the following Aliya, there may be another book for you to follow along, or you may have to look over the asistant Gabbai's shoulder. Please don't crowd him out.
- When the next person is done, smile and shake hands with the assistant gabbai, and/or anyone else up there who looks like they're offering you a handshake.
- In synagogues with the bima up at the front of the room and the rabbi seated up there, go up to the rabbi for a handshake as well. Same goes for the president or chazan, if they're sitting up there. If no one sits up front, or if the Bima is in the middle, don't bother.
- Descend, giving right-of-way to those coming up. One should take a slightly longer than necessary route when leaving the bima on the way back to your seat.
- Greet people on your way back to your seat. Often a smile, nod, and handshake are all that's needed. Depending on the volume level of the synagogue, words might be appropriate too. If someone says "Shabbat Shalom" or "Good Shabbos", you can respond in kind. "Nice job" means just that, and gets a "Thank you." "Yasher Koach", "Shkoyach", "Yeyasher Kochacha", "Yishar Kochacha", or anything that sounds remotely like that, is a Hebrew blessing for your increased strength; it's used as in "nice job, way to go." "Thank you" is a perfectly fine response to this; the formal Hebrew reply is "baruch tihiyeh", "may you be blessed."
- If you used a borrowed Talit (whether a friend's or one belonging to the synagogue), return it now. Someone else might need it, and may take it from you. Otherwise, return it to where you'd gotten it. If it was a communal Talit and it came folded, return the courtesy and fold it before returning it. (If in a Hassidic synagogue or right-wing yeshiva, they may frown on folding a Talit on Shabbat, so just put it back unfolded.)
- In Sephardic synagogues, the Gabbai may be known as a Parnes (rhymes with "car case"). He serves the same role. They may not use the "Ya'amod ..." procedure as above, and you will just start the blessings after being summoned to the bima. You still need your name for the Misheberach.
- In German-ancestry synagogues, the day-of-death anniversary of a close relative will warrant a memorial prayer ("Av HaRachamim") the Shabbat beforehand. This is done after the Misheberach. If you're not sure if this is done in your synagogue, ask ahead of time.
- During the afternoon service (Mincha) on Shabbat, some synagogues require those called to the Torah to wear a talit or jacket; others don't. Chabad synagogues don't require a talit even at morning services. If you're not wearing a Talit, touch the parchment instead with the belt used to bind the scroll. When removed before the reading, it's usually placed at-hand for those called up. If not, ask the assistant Gabbai.