I always recommend the Uno R3 with DIP mega328 as a starter board. There are more tutorials, examples, shields and just about everything for the Uno than any other Arduino. And the chip is replaceable for when you accidentally toast it.
You are likely to to want to do more than you think, and will then need a Mega. So you might as well get it now. It doesn't cost much more, is just as easy to use, and when the time does come when you need one, it is guaranteed to make life easier than trying to squeeze your programme into a Uno.
That is not a requirement I can understand. Are you trying to some HTTP GET POST stuff or is it TCP/UDP? Most arduino compatible modules can do all that but you should be more specific what you need. This feature will cost you something around $50-$80 USD. I have been using the expensive Arduino official wifi shield and have started migrating to the less expensive xbee wifi module + wireless shield.