Definitely. Although there is no such thing as perfect security, a properly configured Linux, *BSD or Unix system can be very secure and orders of magnitude better than any version of windows.
Linux, Unix and *BSD are 100% immune to windows viruses and exploits as long as you don't install WINE ( WINE allows windows software to run on Linux ). There are about 80,000-100,000 windows viruses, while Linux only has about 100, MacOS X 60-100 and Unix about 10-100.
If you use well written firewall rules, SELinux and you configure and use the system properly it can be very hard for anyone to break your security.
Some security experts recommend using only one computer or a secure Virtual Machine (VM) running on top of Linux for your banking and ebay and nothing else. If you use your computer or VM for other tasks you'll open yourself up to a number of threats even if you have a very secure operating system like Linux.
Firefox is also more secure than other browsers, especially internet explorer, although firefox has been targeted due to its popularity.
Tell us about your hardware and what you're trying to do and the OSS experts on the forums will make some suggestions.
Semper Fi