My first response when I read this yesterday was "what? I don't think so." Then I thought about it a little more and continued to read other responses. I have since changed my initial opinion. While I don't think skeet shooting is necessarily a good analog for home defense practice, it will help you gain a lot of knowledge operating and shooting your shotgun, especially since you're very new to them. Skeet shooting should definitely help you improve your hand/eye coordination while teaching you to focus on hitting small moving targets. Plus, any good trigger time will help.
I think it would also be beneficial to get some silhouette type targets, some shells with different shot sizes, different choke tubes if they are an option with your gun, and "pattern" your gun at ranges you expect to encounter inside your home. What I mean by that is to place the target at an expected distance, aim center of mass, and shoot. You'll be able to get an idea how shot size, choke, and distance all affect the spread of the shot. For example, smaller shot, more open choke, and greater distance typically mean more spread.
As to shot size/type. There's a lot of discussion and debate on what works best for defense. It depends on your home construction (layout and materials), proximity to neighbors, and how many people live there, as well as, what you are comfortable with. If over penetration is a concern and distance to target is minimal, small shot such as used for skeet should work fine. I personally have my home defense shotgun loaded with #8 shot. The pellets are small, but, at short apartment distances out of a tight choke, they don't spread much and hit together. I personally try to minimize spread because I am already using undersized shot (If overpenetration/collateral damage weren't a factor, I would prefer to use buckshot.). Keep in mind, real life isn't like the movies; your shotgun isn't going to blow a perfect 1ft diameter hole in a wall from 5ft away. You can miss with a shotgun. This is where the skeet shooting and learning to hit small targets comes back into play. We're generally taught to aim center of mass; however, smaller rounds increase the need for better shot placement (i.e. aiming for the head instead of the chest). Don't get me wrong, shot placement is critical to whatever you're shooting, but a smaller projectile has less room for error.
You brought up steel shot. I personally would avoid it for personal protection. It's just too light in my opinion. You're already reducing shot diameter; so, reducing shot weight additionally just isn't good in my opinion. You have to find a balance of penetrating the bad guy enough while reducing over penetration through the bad guy or a miss.
For the record, I am not nor claim to be an expert. I am simply adding my opinion and thoughts to discussion. What is most important is finding a gun, ammo, and system that works for you and practicing with it.