You don't need to get one of those spray mops with the expensive refills to keep your floors clean. You can get a spray mop that comes with a refillable bottle and reusable pads to mop your floors.
The spray bottle stopped working on the one we purchased, so we just spray the floors with our regular spray bottle, which has a 1:1 ratio of vinegar to water, and mop over it. It works great. We aren't tied down to any company's refills, we can use our own cleaner (which is much healthier, and fragrance free), and it is just as easy.
If you aren't familiar with Plastic Free July, take a look at this post. Basically, it's a month-long challenge to reduce the amount of disposable plastic you use. It's a great starting point if you just want to give cutting out disposable plastic a try.
My first Plastic Free July was in 2013(!), and this month I'd like to highlight some of the things I do to reduce my plastic use, and some struggles. Many things are part of my routine at this point (buying in bulk, refusing plastic silverware, etc.), but I am not perfect, and I always find this challenge to be a good time to refocus and look at what trash I am actually producing.