After reading positive comments, I jumped on the ESP8266 boat.
The ESP8266 chip/board is a WiFi-to-serial adapter (yeah, the "so 70's and 80's" serial port!), which includes a TCP/IP stack and a WiFi Access Point. It can work as a server on your home network, or as an "Internet of Things" (IoT) client, or accept WiFi connections from other devices. Just use your imagination and do some advance planning. All for 5 USD or less.
These boards run at 3.3V, so there is need for a level converter to use them in pair with Arduino and a separate power supply/adapter. Or buy a 3.3V Arduino board (it will run at 8 MHz instead of 16 MHz clock).
There are a few Arduino libraries around, so life is quite easy as long as you understand all limitations involved (for example you will not stream video over a 115200 bit per second channel with a 64 bytes buffer).
My ESP8266 modules remember the AP setting, user and password, so they connect automatically at powerup. Unfortunately they do not auto-start the TCP server, so a microcontroller is needed. Or a new firmware can be flashed ... or developed if you're brave enough!