Season 20 draft--a week from tomorrow, or Monday August 20th. Just FYI though, this season's draft kind of sucks. If you like I can bb-mail you some information about the draft, since you're new and haven't had time to do any scouting. But there really aren't any great players in this draft, and few decent ones by the look of it.
As for picking the players you want to train, you need to pick a skill and a position from your training page, then get your players at least 48 minutes at that position. Hope that helps.