Correction: Spybot Search & Destroy is the full name of the software that you want to download and install. You do not want to download a spybot, that would be deliberately sabotaging your own computer. Spybot bad, Spybot Search & Destroy, good.
I usually recommend Ad-Aware at the same time. Since you run them manually, they're not mutually exclusive.
There's a couple of freeware -- upgradeable to better Norton replacements. Avast is a good one.
Need more details on the trojan in order to help you remove it. What I usually do is google on whatever is reported, and find some manual removal instructions and execute them.
Don't blame Norton -- remember that virus fighting software will always be one step behind the people who write the viruses. Your best bet is to go Mac or Linux, although that just lowers your risk, does not eliminate it completely.