What is difference between 変わり and 変える ? 変える means "to change". Maybe 変わり is just a noun? Or maybe there is some grammar rule?
Japanese has a number of morphologically related transitive-intransitive pairs. The pair of verbs you've discovered, 変わる and 変える, belongs to the largest group, -ar (intransitive) and -e (transitive):
Intransitive Transitive あがる ag-ar-u 'rise' あげる ag-e-ru 'raise' 集まる atum-ar-u 'gather' 集める atum-e-ru 'gather' 溜まる tam-ar-u 'accumulate' 溜める tam-e-ru 'accumulate' 止まる tom-ar-u 'stop' 止める tom-e-ru 'stop'
The romanized versions reflect the actual morphology better than the versions in kana and kanji, but you'll need to recognize the distinction in Japanese spelling, since Japanese isn't usually romanized.
In this case, the root is kaw-, so we end up with the following pair:
変わる kaw-ar-u 'change' 変える ka(w)-e-ru 'change'
In Modern Japanese, /w/ drops out before vowels other than /a/, so we're left with kaeru rather than *kaweru. Historically, however, this root was kap-, and we can see the etymological presence of the original consonant p in both words in the older spellings 変はる and 変へる.
Note that in each -ar / -e pair, the intransitive verb is a consonant-stem verb ending with r-, while the corresponding transitive verb is a vowel-stem verb ending with e-. Consonant- and vowel-stem verbs may be called by another name in your textbook; they're called 五段（ごだん） and 一段（いちだん） in Japanese, and in books for learners they are sometimes called "-u verbs and -ru verbs" or "Group I Verbs and Group II Verbs".
For more information about transitive-intransitive pairs, see Shibatani's 1990 book, The Languages of Japan, pages 235-237. (The chart above is based on the one on page 236.)
The infinitive form of a verb, called the 連用形（れんようけい） in Japanese, is formed by adding -i to consonant-stem verbs and adding nothing at all to vowel-stem verbs:
変わる kawar-u → 変わり kawar-i 変える kae-ru → 変え kae-
This form can be used like a noun, but it has lots of other uses. For example, it's the form that you add the polite auxiliary 〜ます to:
変わる kawar-u → 変わります kawar-i-masu 変える kae-ru → 変えます kae- -masu
For more information about this form, see your textbook. Besides the names given above, it may also be called the continuative form (a translation of the Japanese name), the "Vmasu stem", or sometimes just "the stem". Unfortunately, different people use a lot of different names for the same things.
First, let's change 変わり to 変わる, so we have two verbs. 変わる is an intransitive verb. It means "to change" in the sense of "become different": the subject of the verb is what's being changed. In contrast, 変える is transitive. It means "to change" in the sense of "amend": the subject of the verb is causing the change, and the object of the verb is the thing being changed.
It's the same as the difference between a witch changing into a cat, and a witch changing you into a cat. The first of those is 変わる, and the second is 変える. Japanese has lots of instransitive/causative pairs like this, where the kanji is/are the same, and the rest of the word distinguishes which is which. They can be tricky for native English speakers because we usually use the same word with both meanings (like "change") and use grammatical context (is there a direct object) to figure out whether it's transitive or intransitive. In Japanese, writing that way gives you nonsense sentences.
Often with る verbs, replacing the る with り makes the verb into a noun. 変わり is a change, in the sense of an alteration or difference. There's no corresponding noun 変えり, though.