First off, amateur radio isn't very strict. With few exceptions, nobody is going to be very upset if you miss out on something during a casual contact. If you mess up too badly, you'll be asked to provide the missing piece of information, be it a signal report, location, repeat your call sign because it couldn't be copied, or whatever the issue may be. So don't be too worried about trying and failing. I've had people copy my call sign wrong on occasion too and it isn't a big deal; just grounds for a repeat of it, is about all.
For a non-contest ("rag-chewing") contact, it's largely free-form.
The first step is to announce your presence, or find someone looking to make contact themselves. On HF the standard way to do this is to call CQ ("seek you"); on repeaters practice varies, but it shouldn't be too bad calling CQ there either. You can also tail-end someone else's QSO, calling them up as they finish their contact. Don't make too long a CQ; you want to attract attention, not bore people to death before giving them a chance to respond. Even on HF SSB, a three-by-two CQ (three CQ followed by your call sign twice), possibly repeated twice, is usually plenty enough. If it doesn't attract attention you can always try again, or recognize that the frequency may be in use by a station you cannot hear. Or that simply nobody wants to talk to you, though that's quite rare. A CQ might be "CQ, CQ, CQ, KD8OAS Kilo Delta Eight Oscar Alpha Sierra. CQ, CQ, CQ, KD8OAS Kilo Delta Eight Oscar Alpha Sierra, calling CQ and by". Here, "calling CQ and by" is a fairly common shorthand for "calling CQ and standing by for a call". Don't rush things; someone hearing you may need to tune their antenna before they can answer, for example. Give it at least a minute or two before giving up on the frequency.
Something that does irk me is when people get the order of the call signs wrong. It's "to from from", not the other way around. So if you were to call CL1SGN on amateur radio, it'd be "CL1SGN from KD8OAS", not "KD8OAS to CL1SGN". Some people will actually go so far as to ignore a call if the call signs are given in the wrong order; and with a garbled transmission, you risk mistaking one station for the other.
Assuming the remote station answers your call, unless they know you from before it will usually be a pretty brief transmission just to tell you they heard you and are willing to make a contact. Technically, at this point you have established contact. At that point, you'll usually give the call sign pair again, followed by your first name and location (to whatever precision is called for). Many also give a basic equipment report stating transceiver model and antenna type and (for non-repeater contacts) mention the amount of power they are putting out. If you are able to judge it at this point, also give a RST (signal) report, then hand over to the other station with something like "CL1SGN from KD8OAS, go ahead", at which point the other station will generally respond with the same information. By the time they are finished, you should definitely be able to give an appropriate signal report. Beyond that, the QSO may take off in almost any direction. Just keep in mind the legal limits on what can be discussed over amateur radio, as well as topics that are best not discussed (politics being one of them).
By the time you feel happy with the contact, just courteously conclude it. This doesn't need to be very fancy at all. "Well Jim, it's been nice talking to you. I'll be sending a QSL card via the bureau if that's fine with you. I'm OK in the call book or via the bureau. Have a good evening and maybe we'll talk again some other time. Seven-three and back to you." There are two ways of sending QSL cards, "via bureau" (through the amateur radio association in your country) or "direct" (mailing it directly to the other station). If you want a QSL card "direct", be sure to say so as well as state whether (it better be!) your address is correct in the amateur radio call book. Also, don't expect unsolicited QSLs for contest contacts, and don't give such extraneous details during a contest contact; just send a QSL yourself and write "PSE QSL" ("please QSL") on it. The remote station will likely tell you either that they don't accept QSLs via the bureau or that they are OK in the call book as well. Either way, note it in your log if you want to send a QSL card later.
At this point, it's pretty much up to what you want to do next. As a general rule, it is commonly said that the station that called CQ "owns" the frequency (because they were "there" first), and it's courteous to either disown the frequency ("you can have the frequency, Jim") and/or state that you are going off the air ("KD8OAS going QR-tango", for QRT meaning you are closing down your station). Remember to finish with stating both call signs involved, so that anyone listening in will know who made contact with whom.
Once that's all over, write down the conclusion time for the contact in your log book, grab a blank QSL card from the pile, fill it out, put a stamp on it and place it in your "out" pile. Then, go reward yourself with a cup of hot cocoa or whatever strikes your fancy.
The contents of a contest contact will depend very much on the specific contest, but is generally pared down to the absolute minimum: call sign, signal report (normally a simple "five nine" even if the signal really isn't, to save time) and whatever the contest exchange happens to be; a locator, serial number, or something else. The basic form is the same, only sped up about 10× or so. During a contest, if you are holding a frequency you might not even give your own call sign more often than every half dozen contacts or so. Of course, if you are scanning across the band and working whatever stations you can find, you'll need to give your call sign each time as nobody on the particular frequency will have any idea who you are otherwise. A contest CQ is also much shorter, sometimes shortened as much as to just your own call sign once followed by the word "contest".