The target breach has nothing to do with identity data and everything to do with credit/debit card numbers and security codes being skimmed. The only cards that are at risk are cards that you have used to make purchases at that store during the times that they were vulnerable.
I will leave it to others to discuss recovery from identity information leaks, and I will focus specifically on the electronic skimming attack. These attacks have been known for a while and have been reported on a local level concerning gas stations and poor security on their card readers at gas pumps. When a user would scan their debit cards and enter the pin number, the skimmer would take that data and make ATM withdrawals. Target is a victim here too, we'll have to wait to find out if they were negligent or not.
No matter how you choose to transact your purchases, there will always be some risk with the method. This info applies to transactions inside the US.
Cash
If you make all your purchases with cash you will probably tend to carry it in larger quantities than if you make the majority of your purchases with a card. When you misplace your wallet you can't cancel your federal reserve notes and minimize the loss as you can with plastic. One advantage is that you don't have to go over your card statements looking for mistakes each month if everything is paid in cash. Cash is generally accepted everywhere so it is good to have some on hand for emergencies. I usually carry between $20-$40 with me everywhere. I used to keep an additional $200 hidden in my car, but now I keep it in an emergency backpack in my house with a 72 hour supply of food, water, and other emergency essentials. I have that money in a large variety of denominations because I don't expect anyone to have change (or to admit to having change) during an emergency.
ATM cards
These cards generally require you to enter a pin so that it is unlikely for anyone to be able to withdraw cash without your knowledge. They are vulnerable to the skimming attacks where a new ATM card is created from the information stolen by a transaction on a compromised system. Banks don't always indemnify you against losses through ATM withdrawals, so you could lose a very large amount of money if you are a victim of these attacks. I regard ATM cards to be an unacceptably high risk. I only resort to ATM withdrawals if I have no other payment options and my usual $20 isn't enough for what I need. (I've used one once in the last 3 years.)
Debit cards
Debit cards are a combination of an ATM/credit cards. They can be used for ATM/debit transaction and require a pin and have the same pros and cons as the ATM card above. They additionally provide the convenience of being usable on a credit card network (usually Visa in my experience) but instead of having a credit limit the money is directly withdrawn from a deposit account at your banking institution. They don't always come with the same guarantees as credit cards even when performing "credit" network transactions. Another plus is that you have the convenience of a credit card but less temptation to go into debt if you have no credit line associated with the card.
Credit cards
I could write for pages, but I'm going to try to keep it brief. Credit cards are very safe with anti-fraud measures built-in and limited liability for the user. They are also very lucrative to the issuer not only because they get a percentage of every transaction, but also because people like to hold high balances and pay high interest rates on those balances. Credit cards have a lot of gimmicks to attract users who would otherwise stick with cash or an ATM card. Some of these include a percentage of cash back on purchases or airline miles, or baggage fees waived, and other various schemes to instill both loyalty in a customer and incentive to use their card instead of one of the many others in your wallet. These rewards are worth taking if you have the discipline to pay your balances in full each month and stick to a budget that you have planned in advance. I do the majority of my purchases on one rewards card and I pay the balance in full.
I also discuss major purchases with my wife at the beginning of each month, so I don't fall victim to a lot of impulse buys--but those can hit you no matter the payment method if you don't have a spending plan. Having a large open balance on a credit card may make it easier to do so, so that's an argument against credit cards if you have a weakness for impulse shopping.
On a related note, don't take pictures of your card and share it through instagram/twitter. https://twitter.com/NeedADebitCard
