Self Service Payment machines (for train tickets, gas pumps, motorway tolls, Parking lots) require CHIP+PIN cards. Unfortunately American credit cards and ATM debit cards are (with very few exceptions) either MAGNETIC STRIP+SIGNATURE.or CHIP+SIGNATURE, meaning that you are required to sign the card to an attendant, rather than just enter a PIN number in the machine. Your ATM/Debit card has a PIN but probably doesn't have a CHIP (only a magnetic strip), therefore you won't be able to use it in a machine, because those machines do not read magnetic strips, only chips. In fact you don't even insert the card in totally, but only about 1/3 of it until the Chip is inside the machine.
One way around that is to pay cash at the machine (instead of using credit/debit cards), because those machines accept euro bills. The bank ATM machines will take your Magnetic strip+PIN ATM card, so you can still withdraw cash at the ATMs and then pay cash at the machines. It's not really a big problem because you rarely have to pay more than 100 euro at a machine, whether it's gas (a full tank is generally less than 100€, unless you rent a large SUV), or motorway toll (which rarely is over €50, unless you drive from the Alps to Apulia). Parking is just a few euros, so it won't be a problem to pay parking with cash. The only issue might be trains if you are purchasing multiple tickets on high speed trains, which can be quite a bit of dinero, in which case you might have to withdraw quite a bit of cash before using the self service machines at the train station.
Note that to pay the motorways tolls with cash (rather than cards) you must drive through toll booths that display the big white sign with the cash logo (bills and coins). If the logo shows also a black hand holding the bills, then it's a manned booth with attendant (however they don't take cards, only cash). If there is no black hand holding the bills on the white sign, it means that there is a self service machine that takes cash. Don't go through the toll booths that show the blue CARTE (cards) sign, because those machines take only chip+pin cards.
So, to make a long story short, carry enough cash with you and you'll be able to use any of those self service machines even without chip+PIN cards.