Device that fetches a web API call via wifi showing the result through a display
Take a look at the Microchip Mutimedia Expansion Board. It has a 


Integrated 802.11 module
a 3.2 inch QVGA touchscreen
Full TCP/IP stack and lots of examples, including GET/POST, SNMP, etc.
Headphone and Line out jacks with 24-bit audio codec
5 user-controlled LEDs


It has many other features, but those are the ones you requested. You also need to get a PIC32 Starter Kit. There are 3 types available, a basic (50 dollars US), one with USB (55 dollars US) and one with Ethernet and USB (72 dollars US). They plug into the multimedia board, which is $250.

If you are only building 1, this is probably the easiest route, even if it is over $300.

If you are planning on building lots, then maybe you can get some ideas on how to create your device from this setup.

The Microchip TCP/IP stack is very easy to use and the demo programs they have can be easily modified to do what you are interested in.

Using an Arduino and necessary shields is a good way to go, but this is another option.