Sometimes it may seem safer to go back to the days of the Pony Express rather than deal with the multitude of online threats, but there are some simple steps you can take to minimize the risks you face.
First, get a security suite, although there are about a dozen good suites, free and paid, out there. Go with one that gets good ratings and that you trust, and make sure that you let it update and scan regularly.
Second, I strongly recommend using a password manager that's independent of your browser. Not
that the browser password managers are known to be risky, because they're not. But third-party managers allow you more options, such as mobile device support, that I find essential. LastPass (download) and RoboForm (download) are among the best cross-platform tools out there that will work on both Windows and Mac and have smartphone support.
Third, use two browsers. Do your mission-critical browsing, such as financial transactions, in a separate browser from your "casual Web surfing" browser. Personally, I use Firefox and Chrome. It doesn't matter which you use for which, both are safe and regularly updated browsers. But it's the easiest way to prevent unexpected third-party exploits on otherwise safe sites from snagging your data. LastPass and RoboForm let you keep your passwords at your fingertips, no matter which browser you use.
Bonus tip: This isn't strictly security-related, but I strongly recommend setting up sync if you're using Firefox, Chrome, or Opera. A colleague's computer crashed and he lost, among other things, his browsing data. If he had had sync set up, he could've easily resurrected his tabs, preferences, add-ons, passwords, and bookmarks. Even if you only use one computer, it's worth having all that information backed up.
Obviously, there's no foolproof method for staying safe online, but these tips will help you stay significantly safer.