Firefox Nightly for Android is great in my opinion, I use it on a Samsung Galaxy S2 and it's easy to use and browse websites. The default browser is way too basic for my needs and is childish.
The stable version of Firefox for Android is horrible compared to the new Nightly. Should give it a run.
I also like Dolphin HD and Opera Mini is good for mobile internet plans as it compresses all received data and images therefor saving bandwidth, But if you're just using your home wifi network, Forget Opera, Firefox Nightly and Dolphin are much better.
I'm not sure if the Avast Web guard works with it tho, You won't exactly need it anyway, Just don't install .apk files from un-trusted sources, Check the access permissions the apk has and always check it against VirusTotal.
I refuse to root the Galaxy as it renders the warranty & support useless and breaches the phone contract, 24 month contract so I won't be rooting anytime soon.