Hi Caro68,

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.