Ok now I got my Sony PSP online via our wireless network. The issue from this morning was that I had my router set to work with 802.11g only. The Sony PSP is only 802.11b. So after changing that the PSP went online via the new WPA-PSK support with no problem.

Using the Sony PSP browser for a while, I would say we see a Firmware 2.1 pretty soon. There is a out of memory error coming pretty quickly. Several people reported this issue on discussion boards.
To resolve that the browser cache needs to be switched off. After this change in the settings the cache needs to be cleared, the browser closed, the PSP shut-down and after restart the out of memory error is gone.
But without caching pages are not kept, so this is not a cool situation.

Second issue I found is that after browsing around for a while, the browser seems to stop loading new pages and I have to exit the browser and restart it to fix this.
Besides that the browsing quality is quite good. The user interface is usable. The text input needs a bit of training, but it works.
Google is a charm to use of course. It is fast and most information is displayed without horizontal scrolling.
I made several photos of the Sony PSP browser displaying web sites like Gizmodo, Engadget, Playboy to name a view. You can view them in this gallery.
To learn how to update the Sony PSP firmware 2.0 read this story. The update works for US and Japanese Sony PSPs.