The WEP8266 is cool. And it has lots of fans among "the experts"... who work with it in an expert way.
For those of us with less experience, or who just think that "an Arduino"... WITH easy to use WiFi "on board"... would be useful, there's the Sparkfun "Esp8266 THING".
(The ESP8266 is a CHIP which has some great capabilities. Many sources have built boards around that chip... Some are very "powerful" (have lots of support chips on the board). Some are very cheap. Some are very easy to use. Etc.)
The Sparkfun board is $15. Not the cheapest "ESP8266" (based) module out there, but you're hardly going to have to go the bank for a loan. (I bought 2 on my first order, because so often when getting started with something, I "know" I'm not making a mistake, and start to wonder if the particular instance of the device is duff! When two won't work with my software, I have to concede that maybe it isn't the hardware. And in any case... you get something working... an then, where are you going to do the next project?)
The Sparkfun 8266 offering has been criticised for not having very much memory. "Only" 512k. (My ESP8266 Arduserver webserver used only half of that, and I bet that lots of that was for library subroutines, etc.) More than enough to get started. (You didn't have to buy a Mega to start with non-networked Arduino'ing, did you?)
Not MANY GPIO pins... but, count it ungenerously and you have 7 + an analog input. (WHICH SHOULDN'T BE CONNECTED TO MORE THAT 1.0v, by the way.) (One volt will give a reading of 1023). Include things like the pins used for programming the device, and you "have" more.... but not many. But enough. Just "being upfront" with the fact that it isn't a ProMini with WiFi.
Above all, for your $15, you get a device that is supported by "Getting Started" guides, etc. It Just Works. Doesn't take forever to arrive.
I like it so much, I wrote a webpage to help others have the fun I'm having.
The only hard part of the learing curve is learning about working with TCP/IP generally. The Sparkfun "Thing" is easy to use. (I have no connection with Sparkfun or their "Thing" other than being happy customer, in case you were wondering!)