Are you getting these trojan alerts when you are browsing? What exactly does it say (IE scripshield?File System?) because knowing what type of warning will tell you a lot. For example, if you are getting scriptshield alerts, that means you are visiting a website with a malicious script and every time you visit that site, you will get repeated warnings.
As a new user, I would stick with the default Windows 7 firewall. Third party firewall are mostly to control outbound connections, but if you know what I am talking about or don't care, then you won't need a third party firewall for the most part.
Avast should take care of all your "realtime" protection needs including spyware, but there might be some other antivirus programs you can run beside it that won't interfere with each other, but there isn't an official Avast faq to officially acknowledge which one, so I wouldn't do it if I were you.
However, if you think you are really infected, I wouldn't listen to anybody on the forum including me unless it was one of the experts such as "Essex Boy" who is the visiting malware expert. You can easily cripple your system running scans or running tools that you aren't familiar with.
As for my personal setup or advice, there are a lot of things, but if I only could tell you one thing, then it is to keep your software up to date with sense of urgency. According to this article "According to CSIS, five products are responsible for 99-percent of all malware infections. Many of the targeted applications are vulnerable due to a lack of patching, leaving the user and the network exposed......Java, Adobe Reader / Acrobat, Adobe Flash, Internet Explorer, and QuickTime"
You can run this program
http://secunia.com/products/consumer/psi/ that does it all for you. Or you can disable the plugins that you infrequently use like java, quicktime etc
