The built-in Android browser supplied with the phone does a fine job of browsing, loading sites quickly and handling both mobile and full versions of websites like the Los Angeles Times and the New York Times nicely.
The only real issue is with the predictive zoom, and double tapping on an article or image in the browser will zoom in off center, and then pop to the item you've tapped on. It's a bit jarring, but doesn't really affect the browsing experience.
The worst thing about the browser is that if you are trying to access the address bar by tapping the top of the screen frequently just reloads the page you're on, and you have to tap to the left that menu bar, providing a significant speed bump to browsing.
However, installing Chrome is your best bet. It doesn't exhibit the same zoom issues, and handles tabs in a much more elegant manner than the built-in browser.
Additionally it can sync with your Google account, saving your tabs across a range of devices, and accessing the address bar is much easier in Chrome.
It just feels like a much more solid browsing experience, and if you're already tied into the Google ecosphere, it makes a nice fit.
Oddly enough, the built-in browser supports Flash via an optional install, but not Chrome. Java is supposed to be supported in Chrome, but we couldn't get it to work either and frankly it wasn't worth the fuss to try and figure out what the issue is, beyond it not being an easy implementation on the phone.
As far as connectivity goes, the phone offers 3G/4G LTE options for data on the go, and can connect to Wi-Fi networks of all flavors. Over Wi-Fi, the phone averaged about 6.5 Mbps upload / 2.5 Mbps download over a standard cable connection, and over 3G speeds were 1.22 up and .5 down.
Verizon's 4G network is spotty where we tended to use the phone most, not connecting to the network at all. We did take the unit to San Diego early for Comic-Con, and before the show started it handled 4G very well and easily worked as a hotspot for a tablet and a laptop. That is until the crush of fanboys and fangirls invaded town, grinding the 4G network into a fine, powdery dust, and slowing 3G to a snail's pace.