As TKI suggests, the brush should be used to take off the polish, and then the cloth buffing will convert the leather surface to a hard shine. This should keep any polish from the shoes getting on other things like clothes, or the cuffs/bottoms of trousers. It is also a wise idea to keep the shoes stored away from heat.
Since you have said that your questions are those of a neophyte, I thought the following suggestions might be useful:
High quality shoes are often sold with soft, cotton flannel shoe bags. I have Alden and Allen Edmonds pairs which came with a shoebag for each shoe. They will protect them and your clothes when you are packing them into a suitcase. Also, when they are off your feet, it is a good idea to store all shoes with shoe trees inside them to preserve shape. Both shoebags and shoe trees can be purchased cheaply on eBay or similar venues. I often find them in thrift shops at a couple of dollars a pair!
Scratches and scuffs on shoes can be fixed with a bit of polish and some cloth. One should expect some of this damage with shoes as you wear them, but if they are properly treated, they will simply add to the patina of the shoe. You may also want to invest in good quality cream and wax polishes (they serve different purposes). The best products for shoe cleaning, restoration and maintenance are from a French company called Saphir.
There are several Youtube channels devoted to shoes and shoe care, and you might want to check some of these out.
Kirby Allison, Trenton and Heath and
The Elegant Oxford are three channels that I recommend highly for those interested in shoes. Mr Allison also has an online shop and I have bought shoe supplies from him, including Saphir products and even a deer bone for polishing my cordovan shoes, LOL.
The
Shoegazing blog, maintained by Jesper Ingevaldsson, is an excellent source of information about everything related to shoes. He is Swedish (and the blog used to be only in Swedish), but while English is a second language for him, he is quite good at writing in English, except for the occasional odd usage or grammatical error. There are lots of past articles on every aspect of shoes.
Shoegazing is one of the largest blogs in the world on quality shoes, mainly welted footwear. Articles on shoe care, shoe construction, buying guides and more
shoegazing.com
And lastly, the best book on shoemaking, cordwainers and cobblers I have seen is
Handmade Shoes for Men (1999) by the famed Hungarian shoemaker Laszlo Vass of Budapest and colleagues. It is gorgeously produced, with information on masses of things, and beautiful photographs and illustrations. You can find a review here:
Happy reading and viewing!