I want to add regular expression search capability to my public web page. Other than HTML encoding the output, do I need to do anything to guard against malicious user input
You have to not only worry about the matching itself, but how you do the matching. For example, if your input goes through some sort of eval phase or command substitution on its way to the regular expression engine there could be code that gets executed inside the pattern. Or, if your regular expression syntax allows for embedded commands you have to be wary of that, too. Since you didn't specify the language in your question it's hard to say for sure what all the security implications are.