First, think about your budget and needs. Are you searching for an apartment (appartamento), room in a shared apartment (una camera), or an entire house (casa)?
Secondly, how long do you need it for? The process is quite different for a student looking for a place to live just for one semester compared to someone hoping to stay longer-term, and there are two basic kinds of contract. Long-term contracts are called 'contratti di libero mercato' (free market contracts), and typically last four years with the possibility to renew for a further four. You'll also see them written as '4+4'.
There's also another kind called 'contratti concertati' (mutually agreed contracts) which can be used for three years as a residential contract (uso abitativo) and can usually be extended beyond that. This category also covers temporary use (uso transitorio) which is fixed at a set period of one to eighteen months. In student areas, this also includes student rentals (uso studenti universitari), for between six and 36 months.
The landlord is known as 'il/la locatore' in official language such as contracts, and 'il/la proprietario/a' in everyday speech, while the equivalent terms for tenants are 'il/la conduttore' or 'il/la locatorio/a' (in formal contexts) and 'il/la affittuario/a' (informal).
You might rent directly from the landlord if you respond to adverts online or in newspapers, or you could choose to go through a letting agent (agente immobiliare). Even while searching online, you'll see that some apartments are listed as being rented privately (privato) while others are through an agency (agenzia).
Details about the housing
The term for a studio is 'un monolocale', while 'un bilocale' is a two-room apartment, 'un trilocale' has three rooms and 'un quadrilocale' four.
Sounds simple enough? Be aware that exactly what counts as a ‘room’ might differ from place to place, so one trilocale may have two bedrooms and a living room, while another might include the kitchen in the calculation. You'll also see descriptions explaining exactly how many rooms (locali or vani) the place has, but again, check the floorplan (la pianta del piano) to see what's included, whether it's open-plan, and so on.
If you're hoping to live with flatmates (coinquilini), make sure to know the difference between una camera da letto (a single bedroom) and un posto da letto (one bed in a shared room).
Next, find out if the place is furnished (arredato) or not. If it's described as 'non-arredato', or doesn't specify, look for details about just how unfurnished it is -- are any kitchen appliances or light fittings included, for example? And speaking of the kitchen, note the difference between a cucinotto/angolo cottura (kitchenette, though the former is more separated from the other rooms), una cucina (which may be a large kitchen but may also be too small to sit down and eat) and una cucina abitabile (a kitchen big enough to fit a table inside).
Other things you might have on your wish-list include un ascensore (lift), un balcone (a balcony), una lavastoviglie (dishwasher), un box/garage (garage), una lavatrice (washing machine -- bear in mind that these are still not as ubiquitous as in many countries, with many Italians using launderettes), and un giardino (garden).
Renters in Italy should be on high alert for truffe (scams).
To avoid these, ask to see proof that the person offering to rent to you actually owns the house, and therefore is legally allowed to rent it. You can ask them to show you their visura catastale (property deed), and check the contract thoroughly as well as visiting the apartment and ensuring that it matches up to the description.
Try to avoid paying your rent or deposit in contanti (in cash), and never pay before signing the contract and ideally getting le chiavi (the keys).
The Italian Interior Ministry has an English language guide outlining the proper process legitimate landlords will go through, which you can find here. And if you have any doubts, don't take the risk.
There are two key points here: make sure you have a contract (un contratto) and that you actually read it. You can also ask if the amount is 'trattabile' (negotiable), as this is sometimes the case.
The basic cost you're dealing with is the monthly rent (il canone), but what does it actually include? Check for le utenze (utilities), riscaldamento (heating), spese condominiali (extra service charges), which can vary significantly depending on the area and type of building.
You'll also likely have to pay two extra lump sums when you first sign the contract: a deposit (una cauzione or una caparra) which is typically between one and three months' rent, and an agency fee if you've used one. Make sure to find out the terms and conditions attached to getting your deposit back at the end of the rental period.
Once you're happy with everything, the only thing left to do is sign (firmare) and prepare for the move (il trasloco)!