Note: I am Italian, and travelled my homeland peninsula from north to south, and from west to east.
Sorry to contradict some of the previous answers, but no, aperitivo/apericena are not limited to the Milan area. Maybe it started as a cultural movement in the north of Italy, but I can assure you that nowadays (well, it is since I can remember - that would be at least 25 years) it is widespread across the Italian territory.
How can I find out whether a specific bar offers an aperitivo?
Simple: ask. If you go to aperitivo hours to whatever bar you want, and ask for an aperitivo (from beer to cocktail), they will ask you what you want to drink (you can choose from beer to wine, from analcoholic - try Crodino - to a cocktail), and eventually bring you something to eat (chips, peanuts, etc.)
But I guess you want to know where to find good aperitivoes, with better food to eat.
How can I find out which places in the city I am in offer aperitivi?
Places who offer a decent choice of food to eat with your drink do advertise it. Keep an eye for their signs ("Here aperitivo" for example). Ask any Italian if he/she knows a place to get a good aperitivo, they know for sure.
How much should I expect to pay for the drink / aperitivo?
It is common to pay 6-10€ (depending on the place and on the city). In Rome I'd normally pay 7-8€, in Milan I would expect 10€ or more. In the southern part of Italy I would expect 6-7€.
What should I say when ordering to make sure I actually get this aperitivo deal? (In Italian and/or English)
"I would like an aperitivo, please" - "vorrei un aperitivo per cortesia".
You will be asked for the drink that you want. That's it.
What type of drink should I order to blend in with the locals? Can I order any kind of drink to get the aperitivo?
My favourite question! Yes, you can get whatever you want, but make sure you try at least a "Prosecco" (sparkling dry wine) or a Spritz "Aperol, Prosecco, sparkling water and a slice of orange).
What type of food can I expect?
This varies a lot from place to place. I've seen bars offering only chips and peanuts to others preparing tiny pizzas ("pizzette"), finger food (olive ascolane, mozzarelline, croquettes, chicken wings), or even complex stuff (pieces of pasta-omelettes, salads, dice of chicken with sauce, cold pasta).
What are the aperitivo hours?
Depending on the part of Italy you are, it can be from 17.00 (south) to 19.00 (north). Apericena is also offered during dinnertimes (a little bit later).
Is it an all-you-can-eat? Will I make a bad impression if I take multiple servings? Will someone stop me and scream at me in Italian?
It is mostly all-you-can-eat. No, you want make a bad impression, and most likely no one will mind. Especially by apericena, people are expected to eat more, in order not to have a "cena" (dinner).
Is there a difference between "aperitivo" and "apericena"?
"Aperitivo" implies that you are having dinner later (even if after it most people are so full, that they don't), "Apericena" doesn't. My experience is also that apericena (aperitivo + cena) are offered in a later timespan in comparison to aperitivoes.
How do I avoid tourist traps related to aperitivo?
Ask the locals. I do the same even being an Italian.
What else should I be aware of?
You may be asked to pay in advance. An aperitivo is not shareable (if you are with a friend, and only you order aperitivo, then only you may eat).
