To get started make sure to sweep your floor thoroughly before mopping. Mopping a floor with crumbs still on it can cause the starches in the crumbs to turn to gluten and create spots that are very difficult to clean. Mix together 1/4 cup of white vinegar with a 1 quart of water in your bucket. Mop in small sections and when you need to rinse your mop try to rinse it in the sink before putting it back into your vinegar solution so you are not putting the dirt into the clean solution. Allow the floor to air dry and when it's dry, it should be streak free with no vinegar odor. Be sure that you wring your mop well - standing water can damage wood floors.
White vinegar doesn't smell pretty but it isn't toxic for you. You can add a few drops of essential oils and not only will it make your vinegar smell better it will also increase the disinfection power.