Szczoteczka is a double diminuntive. Szczota --> szczotka --> szczoteczka
Basically, you're not going to be able to form diminutives in Polish until you are EXTREMELY proficient in the language. There is often a change in connotation, and a radical change in meaning, when you form a diminutive (just like in Spanish, only much more complicated). This is not a task for beginners in the language. Learn basic and advanced vocabulary first, get fifty or so books behind your belt, and THEN you can start playing around with diminutives. Precious few foreigners use them naturally. It's a difficult skill that will take you many, many years of hard work to learn. After living in Poland for twelve years and reading A LOT, I still don't have a native feel for them.
I think it depends on your mother tongue. It's easier when you have something to relate to.
In English doesn't exist such thing but in Spanish it does.
We have a lot of diminutives too, and people use them a lot, and depending on the country they will use some more, and some less.
We have terminations such as -ito, -illo, -ino, -cito, -ucho, -uco, -ina, -ete, -ica, and many others that I may be forgetting right now.
I know translating diminutives in Polish into Spanish may be not fully right, but it helps to start and understand.
I guess in English it's more difficult because you just don't have so many diminutive terminations and It's harder to relate them to a meaning or sense. We also have diminutive from diminutives, as the example of szczoteczka (chico, chiquito, chiquitito).
By the way, what I tried to say with ''szczoteczka'' is that when you use the word you just refer to the brush, you are not referring to a ''little brush'', just the brush. I'm not sure if I'm right, but that's what it comes to my mind.
So to end I'll say it's easier to understand when you have something similar in your language, when you don't have it, it's harder because you need to build up a new way to understand the diminutives.
It's just my opinion :)
Thank you so much for your answer! :D