Readiness is managed via the Galaxy at War system and I'll go into the specific ways to earn readiness below:

Single Player

War Assets are the part of Galaxy at War you collect in Single Player. Simply purchase War Assets with in game money to earn varying amounts of readiness, rated by Military Strength. The wiki pages list the Military Strength of the various war assets. Some assets require certain conditions to be earned.

War Assets are not Readiness, but instead their effect (Military Strength) is multiplied by Readiness.

Multiplayer

Playing Multiplayer earns you various rewards that increase Galactic Readiness, such as Promoting a character to level 20 or simply completing missions.

Failure (any mission of any level): 3% Readiness
Bronze Mission (success): 4% Readiness

Mass Effect: Infiltrator

This iOS game will help you increase Galactic Readiness if you link the games. From EA's press release, via Cult of Mac:

Inspired by the deep and immersive Mass Effect 3 universe, in Infiltrator players will help to free prisoners from a hostile Cerberus base and will receive rewards for collecting evidence of Cerberus’ crimes along the way. Every completed rescue and intelligence discovery in Infiltrator will increase a player’s Galactic Readiness rating directly through the Galaxy at War system in Mass Effect 3.

It's now available for Android as well: North America and everywhere else.

Mass Effect Datapad

Datapad is a free iOS app that will allow you to monitor your Galactic Readiness and a minigame which will allow you to slowly increase it. From the Galactic Readiness map you can send select 3 different mission points within each sector which can earn up to a 1% readiness boost for that sector only. It takes anywhere from 1 hour to 6 hours to earn a point. You can increase this by getting various power-ups based on points earned playing.

All of this is based on observations, primarily by users here who have shared their first hand experience.

Galactic Readiness Decay seems to have multiple rates, depending on your readiness score, or perhaps on the source of the score.

In general, it appears to decay at a rate of 0.15% per hour or around 1% per 8 hours, or 3% per day to a minimum of 50%.

There have been some reports of up to 6% per day, though I haven't seen this number echoed elsewhere.

If your GR is low, between say 50-60% then it appears to slow the decay rate to around 1% per day.

It takes very little to beef this back up, if it is causing a problem for you. For example you can play multiplayer, by yourself, and complete the Geth mission ... it's fairly easy to achieve a bronze or silver victory.

As @Edwin notes, if you simply start and exit a game, it will not count as an active day. You have to finish the game, which is why I noted Bronze or Silver being fairly achievable on some missions.