I’ll explain it the way it was taught in the university courses I took in Japanese a decade ago.
There are, broadly speaking, two classes of verbs in Japanese (not including special “exception” verbs like “suru” etc.). We won’t worry about the “official” names for these classes right now. ALL verbs end with a ‘*u’ hiragana in their base forms, so a word can only be a base-form verb if it ends with う、つ、す、ず、む、く、ぐ、ふ、ぶ、ぷ、or る. If it ends in anything else, it can’t be a base-form verb. Class-1 verbs can end with any of the above, while Class-2 verbs can only end with る. This means that when you encounter a -る verb, you’ll just have to determine for yourself whether or not it’s Class-1 or Class-2 by looking it up or based on how the word is conjugated in a given sentence.
To convert a base-form Class-2 verb to its polite form, you just drop the る and replace it with ます. So a Class-2 verb like 見る (miru, to see) becomes 見ます (mimasu) to be polite/formal.
To convert a base-form Class-1 verb to its polite form, you first replace the end verb with its ‘-i’ sibling and then add ます. So a Class-1 verb like 飲む (nomu, to drink) becomes 飲みます (nomimasu) to be polite. Another example, 思う (omou, to believe) would become 思います (omoimasu).
Some very common verbs like する (to do)、いる (to be, person)、ある (to be, non-person) are special class verbs that don’t really follow all the rules that plain Class-1 or Class-2 verbs are supposed to, e.g. the polite form of する is します and you’ll hear that very often in polite Japanese, but they’ll be similar. You just have to roll with it and remember what’s an exception and what isn’t, and in my experience Japanese verbs have far fewer exceptions than English.
A lot of Japanese verbs are also made by just sticking suru onto the end of a noun. So for example, the noun for “study” 勉強 (benkyou) becomes 勉強する (benkyousuru, to study), and you can manipulate the meaning by just conjugating the -する as appropriate.
A nice feature of Japanese verb conjugations is that they are generally the same regardless of whether you’re talking in first-, second-, or third-person, and they don’t change based on singular vs. plural, so there are many fewer conjugations than you have to memorize in English, Spanish, or French. Another thing you might notice is that, if a verb has a kanji, that part almost always stays the same (and usually uses the kun’yomi reading), so all verb manipulations happen by modifying the suffix hiragana.
Hopefully that clears up some of the verb basics you might be wondering about.