Putting together a basic home emergency kit starts with small steps. The Home Depot bucket is a great size and the color is ideal. Store this in one of the corners of your garage near the garage door. In the event of an earthquake, this location will be easier to access.
This is what I have in mine so far. N95 rated face masks. This rating is important as it filters the smallest particles. All weather duct tape is great for many things. 550 paracord is lightweight yet incredibly strong. A folding knife with both smooth and serrated edges can serve many purposes. A traditional pocketknife is another good tool for the kit. The Gerber Dime Multitool is small and compact. The Millennium Falcon multitool offers features the Dime does not. A traditional can opener will be worth its weight in gold when needed. Flashlights with fresh batteries are a must, and I also recommend an LED headlamp. Not all glow sticks are the same, so for emergency purposes the yellow 6" SnapLight by Cyalume are the way to go. A lighter and reliable matches are another couple essentials. Hygiene will be an issue in an emergency and it starts with clean hands. I have a few pairs of both sterile and heavier duty non-sterile rubber gloves. Note that many items are stored in ziplock bags. These bags can also be useful by themselves in an emergency. A portable USB battery charger with built-in cords is a good idea. The LifeStraw is another item that will be priceless in an emergency.
What's not in my emergency bucket is equally important to note. A full size first aid kit is a must. Keep it next to the bucket in the garage. Water, water, and more water. Have plenty of bottled water on hand and simply rotate through your stock every few months replenishing it with newer bottles. Finally, freeze-dried food. There are a lot of good options for pre-packaged freeze-dried food kits. I keep a large bucket from Wise Company on hand next to the rest of my emergency supplies. You're going to need water to use the food, but actual cooking is not required.
Please share additional suggestions in the comments. I'd love to hear what you have in your own emergency kit.