What is the most compatible compression/Codec to in Handbrake use for video that will be played in Chrome and Safari browsers?
If compatibility is your top priority, then you should include two alternative versions of your video on your website, like in this HTML example.

As for the exact formats I would suggest:


H.264 and AAC in MP4: Chrome, Firefox 22+ on Windows, IE9, Safari 3.1
VP8 and Vorbis in WebM: Firefox fallback for Mac and older versions on Windows.
(If you need support for even older browsers (IE6-8), and mobile (Android, WP7), you could add mediaelement.js to your website.)


Make sure to put them in that order to make the fallback work. H.264 or MPEG-4 are recommended because they are the only formats to have hardware acceleration on newer devices. 

Compression settings: start by using constant quality with default settings for video and 128 Kbps VBR audio. For H.264 make sure to not exceed Main Profile and Level 4.0. And since Handbrake doesn't do the WebM format, you could try Miro Video Converter.