I have been using the Avbrief WAP facility on my Nokia 6310i but have recently lost the phone config and was unable to get it working again, so I am now using the YAWS one which is actually a lot nicer.
My Nokia E51 should be arriving soon and that can do a lot more than my 6310i
Apart from tafs/metars, there isn't much one can usefully do on a small screen device. I have an old HP HX4700 PDA which can run in 640x480 and in that resolution it does a reasonable job of rendering web pages with MSLP charts etc - if you are a masochist. Been there, done that, and wasted a lot of time. I would just buy one of the tiny ~£200 "EEE" laptops which have an 8" screen and they do all the website stuff fine, and if you get the WinXP version you can do everything else on it.