While you might be better off sharpening your really nice knives with a water stone, you can easily sharpen your less-expensive knives with a few different grits of sandpaper.
In this video from Paul Sellers' YouTube channel, Sellers demonstrates how to carefully sharpen knives without any fancy equipment. All you need to do is glue a few different grits of sandpaper on to some small blocks of wood. Once you have those made, just grab a cutting board or a book, lay the knife on it with the cutting edge of the blade facing out, and start making circular motions with your sandpaper along the blade. Start with a coarser grit, then gradually move on to finer grits to finish it off. To make the process a little easier, and sure your knife turns out extra sharp, you might also want to use the "sharpie trick".
The sandpaper method probably isn't ideal for extremely nice cutlery, but it's just fine for sharpening utility knives, pocket knives, and dime-a-dozen kitchen knives. Sellers also recommends using some diamond paddles if you don't want to use sandpaper. If you're worried about ruining the blade on your first go, start with a knife you don't care about for practise.
How to Sharpen a Knife with Paul Sellers [YouTube]