Firewall is not included in the free edition of Avast. Yes, you need a firewall and antivirus. There's built-in firewall in Windows XP, Vista and Windows 7 that you can use but if have installed Online Armor, you don't need to use Windows Firewall.
Avast will scan for spyware. You can use on-demand scanner for spyware and other type of malware using any of the following:
Malwarebytes Antimalware (MBAM) http://download.cnet.com/Malwarebytes-Anti-Malware/3000-8022_4-10804572.html
A2 Free (A2) http://download.cnet.com/A-squared-Free/3000-2239_4-10262215.html
SUPERAntispyware (SAS) http://download.cnet.com/SuperAntiSpyware-Free-Edition/3000-8022_4-10523889.html
Avast was only recommending their paid solution but you might want to install WOT http://www.mywot.com/ for IE and Firefox to receive extra protection against fraudulent, phised, malicious and rogue websites.
I've downloaded the free versions of Avast Anti-Virus and Armor Firewall. Do I need both of these or does Avast have a firewall that will do the job so that I don't need Armor? Also, do I need to download separate anti-spyware such as Spybot? Do any of these provide anti-spam/anti-phishing protection? The Avast information says that if you are doing banking online you should upgrade to the paid version, is it true that the free version doesn't provide sufficient protection for these sorts of transactions?