Here are some excellent examples of browsers which support modern web standards:
Using a modern browser provides you the coolest new features, latest security enhancements, and automatic updates. Browsing on Internet Explorer not only diminishes your experience, but also exposes you to security risks, bugs, and issues.
We won't tell you which browser to download. It's your choice. Firefox is a good option, as is Google Chrome. If you're on a Mac, Safari already supports most modern web standards.
Competition in the browser space is a good thing, since it promotes creation of web standards which improve the web experience for everyone. Vendor-specific features shouldn't be generally used by websites implementing modern web standards.
See also: The Ecological Impact of Browser Diversity and Alternative Browser Alliance