If you want to complain but don't want to be THAT person:
pre call: find out what governing body deals with the type of complaint you are making. They ALWAYS have rules. For example, in the UK, the Financial Ombudsman Services have a 7 day acknowledgement stipulation AND if the complaint goes over 8 weeks they need to be advised. If you disagree with the outcome of your complaint you can then take it to them and they look into it for you. It costs the company money to have a decision go to FOS. Normally its around £500-ish.
1: Make a list of things you are unhappy about with the service/situation. Ask if all calls are recorded. Get names and times of your call too. As person above said - NOTES!
2: Tell the person that you would like to make a FORMAL complaint. Not just a complaint, but FORMAL. This will normally get you transferred through to the dedicated complaints team rather than the regular call handlers.
3: List your complaint points. If they try to explain away your points, write those down, but you need to make sure that they have ALL your points.
4: At the end of the call - ask them to read what they have written. It's annoying as hell but it will help you in the long run. Make sure you check you have the name of the agent taking the complaint. They might not be the one who DEALS with it, but names are gold.
5: Ask WHEN you will receive a written acknowledgement of your complaint. Ask for your complaint reference number. If they don't have either of those, ask who the governing body is for your specific type of complaint. If you are calling about an energy bill for example, it should be OFGEM. If its bank charges it should be FOS. If they don't know, then ask to be put through to someone who does.
6: When you get your letter/email of acknowledgement, there should be contact information on there, and a return email/address. Send an email/letter BACK, with your points on there as well. It's a good idea to list them - bullet points or date order or numbered. Attach any relevent paperwork or images (screenshots work really well for phone calls and so on) that support your complaint.
7: They should keep you updated with progress reports on your complaint, but they might not. That's okay - keep ANYTHING they send to you in a dedicated place. Try to respond to requests for evidence or information within 24 hours. They may try to use YOUR delay in getting that information as to why its taking so long.
8: They have a certain amount of time to resolve your complaint. Check online cause it can vary, but with FOS and the FCA its 8 weeks - FROM THE FIRST POINT OF CONTACT i.e: your initial call to complain. If its longer than that they MUST tell you the reason why. At that point you can take the complaint to the higher body, but its normally a good idea to hold off till they uphold or decline the complaint.
9: You CAN dispute the decision and you CAN ask for more money if they offer something too low. Don't accept store credit or a discount on the next bill. Cash Only. If you are cancelling your contract, make sure they are waiving the cancellation fees ON TOP OF your compensation. If they say they can't do that, remind them they are at fault and (if they upheld your complaint) have accepted liability. PUSH BACK on anything you don't agree with. Ask for SPECIFIC terms and conditions. Ask what steps they have taken to stop this happening again.
10: Get your decision in writing, with the T&C's listed. I always check my paperwork and contract I was sent last (if your policy renews they can change the T&C's so always use the most recent one) and check the T&C's match. if they don't - even if its the page number that is wrong - go back to them and dispute it. If they are working from a copy of the T&C's you DONT have, ask when they sent those to you, and ask for proof of sending it. If they are working of a set they never sent you - this is a breach of the contract YOU have, and it WILL be accepted by FOS or whatever.
Now I know this seems like a lot of things, but if you feel strongly enough to complain, you should be treated fairly and lawfully. Learning how to complain is a skill, and you cannot let your concern for being 'that person' override your rights as a consumer. LOTS of companies are getting away with shit because its 'too much work' to go through the process.
PRO MOVES: Be polite. Be respectful. Keeping a log of who you talked to and when. If they say that they don't have notes, advise them that you were told calls are recorded, so they can pull that call and listen to it. If the calls aren't recorded, tell them you'd like to add the lack of correct notes to the complaint - give them the time, date and name of who you spoke to, and what you said. NOTES NOTES NOTES.
I've worked in complaints for a regulated company for 5 years, and at the company for 10 and it's MY JOB to know this stuff, so I hope it helps someone.