In order to conjugate all u-verbs and ru-verbs into their respective polite forms, we will first learn about the stem of verbs. This is often called the masu-stem in Japanese textbooks but we will call it just the stem because it is used in many more conjugations than just its masu-form. The stem is really great because it’s very easy to produce and is useful in many different types of grammar.

Rules for extracting the stem of verbs
ru-verbs - Remove the 「る」
例) 食べる → 食べ
u-verbs - The last vowel sound changes from an / u / vowel sound to an / i / vowel sound.
例) 泳ぐ → 泳ぎ
Exceptions - 「する」 becomes 「し」 and 「くる」 becomes 「き」.

Other verbs are also sometimes attached to the stem to create new verbs. For example, when 「出す」 is attached to the stem of 「走る」, which is 「走り」, you get 「走り出す」 meaning “to break out into a run”. Other examples include 「切り替える」, which means “to switch over to something else”, and 「付け加える」, which means “to add something by attaching it”. You can see how the separate meanings of the two verbs are combined to create the new combined verb. For example, 「言い出す」 means “to start talking”, combining the meaning, “to speak” and “to bring out”. There are no general rules here, you need to just memorize these combined verbs as separate verbs in their own right.
Learn Japanese Verbs
ru-Verbs and u-Verbs
Negating Verbs
Transitive & Intransitive Verbs
Stem Of Verbs
Yodan Verbs
Common Yodan Verbs
Ichidan Verbs
Common Ichidan Verbs