The key difference between the US bank's "chip & signature" and the "chip & pin cards used in Europe is the verification aspect when you make a purchase. Both cards have an embedded microchip which carries the necessary data for the credit card and user, much the same way the older magnetic stripes did.
With a "chip & pin" card, once the card is inserted into the credit card terminal, the customer has to enter their PIN to verify they are the authorized user of that card. The terminal then prints out only a receipt.
With the the US "chip & signature" cards, there is no PIN required, rather the terminal prints out a signature slip, same as credit card systems have done in the past, for the customer to sign that they are the authorized user.
Fortunately the majority of credit card machines recognize and accept both kinds of chip cards. And will automatically handle the card the correct way, which may surprise the occasional clerk in Europe or Canada, when their machine suddenly prints out a slip to be signed. (I have gotten into the habit of asking, if by chance the clerk doesn't hand me a slip to be signed, saves them grief from accounting later on ;-)
The only places that the US card are a pita is at automated kiosks that sell things like bus tickets or movie tickets. Those machines only accept chip & pin cards as a security measure.
As far as which bank, most US banks offer "chip & signature" cards, though staff at smaller branches may not know about them as there is little demand. I hear rumors that some banks have "chip & pin", but I have never been able to find one (I think many folks get a chip & signature card with an ATM pin and then assume it is a chip & pin, but they don't work that way).
I have both credit cards and check debit cards from a variety of US financial institutions, that are all chip & signature and use them all over the world (literally) without issue.
Tips about ATMs and cash advances, bring a check card / debit card either Visa or MasterCard to use for getting cash from your account. A credit card will charge you interest from the moment you take the cash, irregardless of whether you pay it off or not before the due date. A debit card simply deducts the cash from your account. Most major US bank debit cards now carry the same level of protection against fraud as do credit cards.
Also getting a cash advance inside the bank instead of withdrawing through the ATM may avoid ATM fees by both your hometown bank (they treat is as a purchase) and the withdrawing bank. It is a loophole that Visa/MC haven't plugged yet ;-)