Well, the bank I like is Leon and Santa Cruz. Both allow you to have accounts in USD and RD pesos, have online banking and allow you to transfer easily between the accounts. Most banks in the DR have online banking nowadays. Popular also allows you to pay many bills online, however I do not like Popular at all because there are always long lines in all branches. leon offers prepaid VISA card which works with Paypal and is great for online purchases and protection (you transfer money on the card via online banking right before you make purchase).
Most banks in the DR work via US corresponding banks and it takes 3-5 business days to transfer money form abroad to the DR account. Two of those 3-5 days is the time the DR bank "holds" your money. Also you must transfer less than 10k USD or equivalent in foreign currency because 10k and over your transfer will be held up and proof of legal origin of funds will be requested from you and/or your sending bank.
Most of the DR bank offers ATM/debit cards, but only few offer truly VISA debit cards for making PIN-less (swipe and sign) purchases. Among those are Leon, BHD and Santa Cruz. Other banks offer at-most "Maestro" branded PIN-based debit cards. Most cards can be used abroad with currency exchange rate at market rate + 3%, ATM withdrawals are possible with usually a US$5 fee per transaction.
Most internet banking allows you to set up pre-approved payees within the DR for making local wire transfers. These must be set up through the local branch. Most DR banks now makes due diligence on their foreign customers who do not have cedula and be prepared for 5-30 day wait. Foreign bank references usually help. Also, exact procedures vary by branch within the same bank, as some branches may be less strict than others on what are official bank policies and procedures (some branches of the same bank may open you an account while waiting for due diligence to come through, while other branches of the same bank may put your file "on hold").