At the start of the series, Westeros is still enjoying an unusually long Summer. The pilot episode offers up a few solid dates that will help sort out the timeline later. For instance, Sansa announces she's 13 when she's making her argument to her mom that she's totally old enough to be thinking about marrying Prince Joffrey. King Robert also helpfully notes it's been 17 years since his rebellion against the Targaryens and the death of his beloved Lyanna Stark. Across the Narrow Sea, Daenerys marries Khal Drogo in the premiere and discovers she's pregnant in the season's third episode.
Daenerys's pregnancy is the best way to mark how much time passes in season one. In the penultimate hour, she appears to be full-term when she gives birth to her stillborn son. Add in the weeks it takes for her to notice she's pregnant, and it's safe to say roughly a year passes between the beginning of the series and the final episode of season one, when Robb kicks off the War of the Five Kings.