Many of the drafted players will be signed by today or the end of the weekend. Teams will then l have them to the Spring training facility for orientation/physicals/and a mini camp.
That is usually 4-5 days, maximum. For most players, the team already knows where they will be assigned. For a few players, that might change based on how they perform in the mini-camp. Games begin on June 20; the players will head to their assignment around June 16/17 so they have a day to find places to live, etc.
For assignments between short season A and Rookie ball, while most high school grads will be in Rookie, there will be a large number of college signees there also.
How the assignments between short season A and rookie ball certainly in part depends on draft position, but also maturity, and organizational philosophy. Long way of saying the criteria for those decisions is known only to the club.
Pitchers who are high picks with a lot of innings in the Spring will be handled much differently. Even if they sign, they may not get an immediate assignment, and when they do, they likely will be on a 50-60 pitch count for the rest of the reason.