Since I've actually measured decomposition rates before, I can tell you how I did it, but it wasn't easy.
Decomposition is the process of recycling living (organic) material back into the non-living (inorganic) components it came from originally. All living things (with the exception of animals living on the bottom of the seafloor at hydrothermal vents) ultimately rely on plant material to grow and make new organic matter. Plants use carbon dioxide, water and nutrients to make organic compounds. So, organic garbage (food waste, for example) will break back down into carbon dioxide, water and nutrients. This takes a long time, and there are many, many chemical reactions that have to take place for this to happen. These chemical reactions are helped along by decomposers: bacteria, fungi, insects like millipedes, etc. Some of the organic garbage will be converted into new organic matter (more bacteria, fungi and insects), some will be released as carbon dioxide and water vapor, and some will run out as liquid water. The water will contain lots of water-soluble nutrients (nitrogen, phosphorus, potassium, calcium, etc.). Some stuff that is really hard to degrade will be left behind. This can include things like walnut shells, bone, etc.
For my set-up, I took some dead plant material ("litter"), carefully dried it, weighed it, recorded the "initial" weight, and then cut it up into small pieces that were roughly the same size. Then I sewed these pieces into some nylon mesh, making a round tea bag filled with litter. The mesh was large enough to allow bacteria and fungi to enter, but small enough to keep the litter inside the bag, so I could cut it open and re-weight the litter later. (You need a sensitive scale for this.) I sandwiched the mesh bag of plant litter between two layers of soil in the bottom third of a large mason jar (like the kind folks use for canning vegetables). Then I misted the soil with water (bacteria and fungi need moist soil to grow), covered the jar with Saran Wrap held in place with a rubber band and left the jar in a dark room at a constant temperature (about 70 degreed F) for three to four weeks. I made sure to mist the soil every few days.
Now, to do a good scientific study, you need to compare two or more treatments, with each treatment having several identical replicates, so I had more than just one jar. I had a treatment with just soil (no plant litter) as a control, and five other treatments with leaf litter from five different species. Each of these six treatments had five replicate jars. I was very careful to make sure ALL the jars were as similar as possible in terms of the amount of litter, the amount of soil, the type of soil (I mixed a large container of soil beforehand), the type of jar, the temperature they were kept at, the time they were left to decompose, etc. The only thing I wanted to vary in my study was the type of litter itself.
When I was ready to measure decomposition, I measured a lot of things, in a very complex set of analyses. I was trying to track the organic carbon in the litter and see exactly where it went (how much went into the soil, how much went into the air as carbon dioxide, how much went into bacteria and fungi and how much remained). What you can do is just re-open the mesh bages, re-dry the litter, and then re-weigh it. The difference is the amount of plant material that decomposed. Again, you'd need a very sensitive scale to measure the small differences, and you'd need to keep track of the initial and final weights of litter in all the jars to compare the loss of plant material in each one. It sounds like what you want to compare is soil type and not litter type, so you'd need to use different soil types but the same type of litter. Use something that you know will degrade quickly (grass would work great). And be sure to mix the soil well before you put it into the jars, so that each replicate is as similar to the other replicates as possible.