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One of the biggest realistic threats for people after a breach is collateral damage. A disappointing number of people use the exact same password for everything. Since a number of websites don't take your credit card information or any other vital information, a breach can seem minor. Except, the hacker will likely have the email address that you used on the site and a password. Please make sure that your password is not shared.
This is actually one of the most important parts. If you have a shared password, you'll need to go through your list of important sites and start changing passwords. If someone has your email address and the password for your email, then they can control everything you use online. Even if you use separate passwords for your shopping and banking, they can just use the lost password function to have a new one sent to your email address, if they have control over it.
Go through all of the big sites, especially anything that could be related. Check Amazon, iTunes, PayPal, banking sites, any major shopping site, forums, Facebook, Twitter, etc. If you have to do so, you can do a search of your email address for “billing” or “account information” to track down extra accounts tied to email.
If you need help keeping track of all these passwords you're creating, I strongly suggest using something like KeePass. This database lets you just come up with one very strong password to protect a list of many other passwords. You can just press a button to generate a random long password, and the copy and paste it as necessary.
Note that secret questions can fall under the same umbrella. If you use the same question, and they gained access to it, then it may be compromised on other sites. Adjust this as necessary to ensure that they can't use the "lost password" option to get access to your account.