Scared of using the internet

[Yuhaney]

Am I being too irrational?

Not necessarily. It shows that you understand that there are some risks involved when dealing with the web. Just be very careful around suspicious links or attachments and you'd most likely do just fine. As a rule of thumb, if you're ever unsure then don't open.

I used to use a PC and seemed to get all sorts of nasty things constantly, but it's been okay so far now that I switched to Mac.

Don't get too comfy with the idea since no OS is safe from malware and viruses. As Romulator said, just get yourself a protection and do scans regularly. Doing comparisons will help you decide which software you'd like the best.