Not much time at all. Apparently less than a month. The first season closes with Dany hatching her dragons, and the second season opens with Tyrion assuming his position as Hand of the King. According to this fan-created spreadsheet (WARNING major spoilers for viewers who haven't read the books) the hatching happened in the first month of year 299 AL, 20th day. Tyrion's stint as Hand started in the second month of the same year, 10th day.
As for the seemingly fast progression in the characters you mentioned, we need to remember that the show is an adaptation of a series of seriously thick books. There are a lot of story details that never get mentioned in the show due to time/budget constraints. For instance, Bran's new found freedom of movement using Hodor's back pack and his specially saddled pony galvanize his efforts to lead a new life.