King Alex:
Either Mozilla (the Mozilla suite browser or the Firefox variant) or Internet Explorer are flawless. I would recommend mozilla's rendering engine over that of IE (meaning that Mozilla browsers render pages the right way, while IE does not.
Regarding security, I also find mozilla browsers to be more secure than IE.
However, there are measures to deal with to protect both:
First of all - set your browser to deny all third party cookies. That goes for both Mozilla and IE.
For IE:
You can try this one out. It is basically a filter that gets rid of all nasty domains. It is very strong however, so I recommend you read up on what it does before installing it.
-->
https://netfiles.uiuc.edu/ehowes/www/resource6.htm
Another resource to look at:
This website holds good information on various browsing related / internet related security issues, and have good links to external resources as well.
http://www.mvps.org/winhelp2002/nopopups.htm
In general, try to repair IE, but use a spyware tool like Spybot S&D or ad-aware SE to fix your specific problem
www.spybot.info and
www.lavasoftusa.com
also try:
http://www.javacoolsoftware.com/index.html and download Spywareblaster and SpywareGuard.
This has been repeated infinitely, and all the above tools are very good.
In regards to your question on mozilla and security:
If configured correctly, either browser should represent much of a security hazard. You need to limit the damage that can be caused by:
-javascript
-activeX
-other scripts
Look at your settings for Internet zone, Thrusted zone and Restricted zone.
In all of them, disable:
-downloading and running UNSIGNED activeX applets.
In restricted zone: disable ANY activeX (there is a reason why they call that zone restricted

)
Also disable all script functions, and navigation across domains.
In internet zone:
disable automatic confirmation for downloads!
deactivate active scripting!
set security level for "software channel" to high
disable allowing windows opened by scripts to run without a set size
Note that all of the above IE settings are rough translations from my Norwegian build. I hope I made sense though
