Here is what I found on the Zune site regarding the browser usage:
You can surf the Internet on your Zune HD, as long as you're able to connect to a wireless network. Connect to a wireless network on your Zune HD 1. From the home menu, touch internet. The browser will automatically launch. 2. Follow the instructions to connect to a wireless network. After you connect, the browser will navigate to the last page you visited. • To search, touch the search icon in the lower right. A keyboard will pop up. Type in your search terms and touch search. • Touch the favorites icon to see your favorites, or add a site to your favorites list. • To navigate to a specific web site, touch the three dots in the lower bar to display the URL input bar. Touch the current URL to bring up the keyboard and enter in the URL. Then touch go. To exit and return to the main menu, press the home button.
The QWERTY keyboard that appears helps you enter URLs by having a .COM and / button readily available and it is easier than using the keyboard on the PSP. There is no predictive text, but your history is filtered as you enter a URL in case you have been to the site before. You can also manage Favorites in the browser. As you can see in my video above, you cannot view YouTube videos in the browser and you will find it is limiting when trying to access all your favorite sites.
I think the browser is something that should be able to be updated via a firmware release and I do hope it is improved upon over time because the hardware supports a better browsing experience, even if most people just want to use their Zune HD for music and video enjoyment.