A "home" "medical" kit is an extended first aid kit. If you are at home, you can phone 911, so sometimes first aid isn't even required for those few minutes until the EMT arrives at your door.
2"x2" and 5"x5" gauze pads
1 elastic adhesive bandage
1 clear tape (the brown one irritates my skin)
waterproof band-aid (different sizes)
emergency eye wash for chemicals (there's one for physical injuries, I'd advise against it)
+aspirin, benadryl, liquid active charcoal, imodium instant etc.
It's a good idea to retake FA classes every couple years, because the opinion about some methods can change. When I took my first FA class 18 years ago, I was told to apply compression to the armpit for a bleeding arm, or the groin for a bleeding leg, to pinch those arteries, reducing blood flow. Today it's not advised and they tell you to just simply raise the bleeding area above the heart after applying a pressure bandage.