This is a website about hacks, so you know I'm not going to advocate for a cleaning product unless it really gets the job done. Lifehacker has years' worth of archives explaining how, for the most part, common things like hydrogen peroxide, baking soda, and vinegar get the major cleaning jobs taken care of, and all you need is a rag or a sponge to go with them.
Still, there are some more specialized products that are worth having on hand. These are what I use to make cleaning easier and faster.
Useful cleaning tools you might not have
First up, you need a little squeegee. In fact, I think you should have a few. The first one should be stored in your shower and, after you finish washing up, you should use it to get the fog, scum, and water off your shower doors and walls. Doing this every time you bathe takes just a few seconds, but drastically reduces how much and how often you need to clean the tile and glass in the shower. Get one with an adhesive holder so it dries straight up and down, then stick it to the shower wall. This one is $9.59.
Second, you should keep one in the kitchen. Use it to scrape wet messes off your countertops or to gather spills on your floor. Why waste a bunch of paper towels or dirty up a rag when you could just push the wet mess off the counter and into a container, then put it in the trash? Pick up a two-pack for $9.98.
Next, consider some soap-dispensing brushes. You may have seen these for dishwashing, which is great, but they're also perfect for scrubbing floors, tile, and hard-to-reach spots, all without having to fill up a whole container of soapy water and drag it around. Obviously, keep the one you use for household cleaning away from the one you use for the dishes, so get at least two. Better yet, this one for $8.44 comes with detachable heads, so you only need one handle.
Speaking of brushes, it's time to upgrade to an electric one. Stop getting on your hands and knees, putting too much elbow grease in, or over-exerting yourself when you could outsource the strenuous part of your labor to technology. You can get an electric spin scrubber with a telescopic handle and eight different brush heads for $39.99, then use it as a handheld or long-reach solution when you just don't want to scrub.