(Warning, Spoilers below.)
At the end of BioShock, the player (Jack) partially becomes a Big Daddy by reconfiguring his larynx, spraying on some Big Daddy pheromones & outfitting himself with a Rosie's suit. However, you do not get a Drill or Rivet Gun. This process is necessary so that Jack can get a Little Sister to open the door for him that will let him reach Frank Fontaine.
Once you have donned all the needed components to make a Little Sister think you are a Big Daddy, you are assigned to protect a Little Sister as she leads you through a series of corridors called the "Proving Grounds", which were originally meant as a trial to prepare teams of Big Daddies & Little Sisters for attacks by Splicers. During this escort mission the Little Sister will periodically stop at corpses to extract ADAM. At those points you must defend her from attacking Splicers. There are no penalties if the Little Sister dies, but Dr. Tenenbaum will rebuke you for it & tell you to get another one from the vents.
In BioShock 2, the player character actually is a Big Daddy- the fourth Alpha Series Big Daddy to be created, named Subject Delta. You are outfitted with the basic tools that it takes to be a Big Daddy, such as a Drill, Rivet Gun & suit. This suit is less bulky than the Bouncer's suit, allowing better manoeuvrability. There are drawbacks, though, such as lower endurance. Find out more about this on the page about Subject Delta.