If you haven't already, take a look at Blynk.cc's website. Blynk provides a (drag-and-drop configurable!) smartphone/tablet app and a number of skeleton programs for many platforms and interfaces including, of course, the Arduinos.
In brief, your Arduino program can report data to their server, and the smartphone app can read and display that, as well as send control signals back to your board. They make it simple to accomplish basic things and quite easy to expand it to do more. The ESP8266 is a good choice for a web interface, especially if you're using an Arduino Uno or other Atmega328p-based platform, because it takes about the least amount of the MCU's limited resources of any of the compatible network interfaces.
Because your Arduino program will be configured as a network client you will avoid the hassles and potential security issues of configuring your local network to allow access to internal server from outside. The Arduino sketch opens an outboud connection to their server, just as a web-browser would, so you won't need to "poke holes" in your router's firewall to let in your client and anyone else that wants to mess with your stuff.
(Note: I have no connection with them, but I think they have a strong platform for Internet of Things hobbyists and I use their stuff.)
Blynk supports Arduino WiFi, WiFi 101, CC3000, and ESP8266 w/original (AT) firmware. This device (Atmega328p with ESP8266) is logging temperatures from 3 DS18b20 thermo sensors to Blynk as I type this: