The first transfer period will start immediately when you start a new career and you will also receive an e-mail about it. For it to unlock, you must sign a player on Deadline Day. An easy way to do this is to play with a team with a lot of money (I did it with Barcelona) and then, one or two days before deadline day, offer a huge amount of money for a player with little skill. Right before or during the first period of Deadline Day the player will inform you he accepts. Now, during Deadline Day, send him an unrefusable contract offer to make him sign.
Deadline Day starts with 10 hours left on the clock. Each time you choose Continue, time will decrease with 1 hour. Don't panic if the e-mail says you'll receive a respond within a few days. On Deadline Day, you receive are respond every 1 or 2 hours.