Train Italy

No, you don't need to buy the tickets in advance, but it will be a lot cheaper if you do buy them in advance. When you buy tickets (either in advance or on the day), the tickets will be for a specific train and will give the date time, train number, coach and seat numbers on the ticket. You are then committed, you cannot change your booking.
You buy the tickets from the company that runs the trains. Either in advance from their website: http://www.trenitalia.com/ or on the day at the station. If you buy on the website, choose a "print-at-home" ticket. You print this out and take it with you, it is all you need.

There are three levels of fares - Super Economy, Economy, and Base. The SE and E are deep discount tickets but very limited in numbers. Once sold there are no more. The tickets also come with restrictions as to changes and refund - generally none. Buying online and early will allow you to take advantage of the lower prices IF they are available but you are then locked into a special train, day and time. If you are concerned about the train selling out - rarely if ever happens.

gabriela,

No, you don't need to buy tickets in advance as you can easily buy them when you arrive in Italy. The only reason to buy tickets in advance is to save money on the high speed trains. There are no discounts for the Regionale trains, so tickets for those are best purchased in Italy, preferably at least the day prior to departure. However, as those trains have compulsory seat reservations, you MUST be willing to commit to a specific train, date and departure time. Note that the cheapest Super Economy tickets are non-refundable and non-changeable once purchased, so choose carefully.

If you decide to buy advance tickets, you'll need to know which stations you'll be using. A few thoughts on the trips you'll be making.....

• Milano Centrale to Venezia S. Lucia - you'll be travelling via high speed train, and you can travel either with Trenitalia or Italo. However be sure to note the stations used by Italo, as in some cases they depart from different stations in Milan. You can buy tickets for either rail line either on their websites or from www.captaintrain.com (no difference in price).
• Venezia S. Lucia to Cinque Terre - on that route you'll be using a combination of high speed and Regionale trains, and you can choose routes either through Milan or through Florence (I find the Florence route a bit quicker). One VERY important point to mention is that with locally purchased Regionale tickets, you MUST validate them prior to boarding the train on the day of travel, or risk hefty fines which will be collected on the spot!

Check the Trenitalia website for the various travel solutions from Venice to the C.T. In planning your route, you'll need to specify which of the five towns you'll be going to.

Two more points.
1. Don't buy the Milan-Venice tickets if you are arriving at Milan airport that day. Your flight may be delayed, or have unusually long time clearing immigration and getting your luggage. If you buy advance nonrefundable tickets and you miss the specific train, the tickets become worthless and you will need to buy a new full fare ticket. The advance purchase discount isn't enough to warrant the risk IMO.
2. There is also another company operating high speed trains, Italotreno. They do not have direct Milan-Venice trains, but Turin-Milan-Bologna-Florence-Rome-Naples-Salerno and Venice-Bologna-Florence-Rome. Their prices, pricing structure, speed is much the same as Trenitalia, but fewer trains. Some people say (no first hand knowledge) they are nicer trains. http://www.italotreno.it/en

One other reason to purchase advance tickets...

If you are traveling in a large-ish group and it is very important to be seated together, purchasing your tickets in advance is warranted. Waiting until the day-of, you may be split up in different carriages which may be difficult if traveling with children or elderly family members.

I have only traveled to Italy in October and November so I don't know just how full the trains (on popular tourist routes) get at the height of the season. I have seen some pretty full trains departing Venice in October, however.

Trenitalia is the state-run train while Italo is a relative new-comer private company. I have traveled on both and find them compatible in all aspects. I would, however, purchase your tickets via their web sites versus a third party. Just remember to use the Italian names for cities (Milano and Venezia) and Google the main stations before doing your search (Google Maps works great for this).

If purchasing your train tickets at the station, you WILL need a credit card with a chip and you will need to know your PIN. (If you haven't been issued a PIN to go with your new Chipped cards, factor at least 10 business days to get one from the issuer.) I used both true Pin & Chip and the American style Pin & Signature cards in Trenitalia machines at the stations. But, American debit cards and credit cards without chips will not work. (And ticket agents tend to frown and grumble if they have to run a credit card the old fashioned way, particularly if there is a line of people waiting.)

One addition to the above post:

If purchasing your train tickets at the station, you WILL need a
credit card with a chip and you will need to know your PIN

Tickets may also be purchased with cash (euros).