There are multiple verbs in Japanese that mean “start doing something”, working like auxiliary verbs, and with slightly different usages. They all follow 連用形 of a main verb of action.
始める（はじめる）
始める is the “standard”. Compared with 出す, 始める is slightly more literary. Also 始める is preferred over 出す when the action/event is gradual, long-lasting and/or intentional (more on this in 出す section below)
- 雨が降り始めた -> It started raining
- 彼女はピアノを習い始めた -> She started taking piano lessons
- うちの子供が固形物を食べ始めた -> My baby started eating solid foods
Note 始まる is never combined with another verb like 始める
出す（だす）
出す is almost always interchangeable with 始める. However 出す is slightly more colloquial than 始める, and often emphasizes the action/event is sudden, unexpected and/or out of control.
- 突然警報が鳴り出した -> Suddenly the alarm started ringing
- 彼は急に走り出した -> He suddenly started running
- 急に雹が降り出した -> Suddenly hail started falling, it started hailing
Again 始める and 出す are mostly interchangeable, but 始める is used more for gradual, lasting, intentional actions/events, while 出す is used more for sudden, unexpected, unruly and annoying actions/events. See the below examples that illustrate this. It’s not wrong to used the other one, but it will be a little unnatural:
- 私はスペイン語を勉強し始めた(
出した) -> I’ve started learning Spanish - 三月になって少し暖かくなり始めた(
出した) -> It became March and it started warming up a little - 患者の容態が少しずつ回復し始めた(
出した) -> The condition of the patient has started improving little by little - 彼女は大声で泣き出した(
始めた) -> She started bawling - 彼は急に怒り出した(
始めた) -> He suddenly started getting angry
- うちの子供が歩き始めた(
出した) -> My baby started walking (learn to walk) - 彼は何も言わずに歩き出した(
始めた) -> He started walking without saying anything (started to walk from stand still)
掛かる・掛ける（かかる・かける）
掛ける・掛かる are always used to describe the action has just started, and often gets interrupted.
- 彼は何かを言い掛けて黙った -> He started saying something and then shut up
- 彼は御飯を食べ掛けたが、すぐに席を立った -> He started having meal, but then left the table right away
- 彼は一旦立ち上がり掛けたが、また座った -> He started to standing up, and then sat back down