Sparkfun has a pretty comprehensive weather meter package that sounds like what you want here and as for tweeting it depends on where it is if you have WiFi you can get a wifi shield or if you have ethernet and ethernet shield or if none of these are available you can use a cell phone data connection.

Get yourself an Arduino board + ethernet shield. You'll need sensors (temperature, wind, humidity,...) aswell. You'll find a huge selection in your preffered online tech store. Then, learn some low level programming on Arduino, involving the ethernet library, code a small webclient and off you go. It's not rocket science ;)