How to use ESP8266 wifi shield as a hotspot?
Connecting through the ESP8266 to the internet would require the ESP8266 to handle Network Address Traversal and/or routing, which is not officially supported.

However, if you are prepared to do a lot of work, it can be done. The performance and reliability is bad, but it can be done. See http://bbs.espressif.com/viewtopic.php?t=960 Note that this will require far more skills than running the examples in the Arduino IDE.