始める|はじめる

1. 1Marks the moment an action gets under way.Ex. 1
2. 2Marks the start of an ongoing or habitual activity from some point in time.Ex. 2

Neutral and common in both speech and writing. It can describe a single action getting under way or the start of a routine, and it also works with non-volitional events such as rain starting to fall. Compared with 出す, it sounds neutral rather than sudden. Compared with ようになる, it marks the starting point of an action rather than a resulting change in ability or habit.

• 出すN4「出す」 attaches the same way and also expresses starting, but it highlights the momentum of the moment something begins, often with a sudden feel, and is frequently paired with 「急に」; 「始める」 is more neutral and also suits actions that unfold gradually or continue. For things like crying or rain starting to fall, the two are often interchangeable, though the nuance differs.
• 続けるN4「続ける」 means an action keeps going or that you keep at it, focusing on the ongoing process and its duration, while 「始める」 looks only at the point of starting. Both attach to the ます-stem (the ます form with ます dropped) and are neutral in register, but their meanings differ, so they cannot be swapped.
• 終わるN4「終わる」 is the opposite of 「始める」: it indicates that a continuing action has been carried through to the end ('finished doing'), so the two are not interchangeable. The attachment is the same (both take the ます-stem); the more formal, written alternative is 「終える」, and the pattern often pairs with 「まで」 or 「たら」 to mark a time limit.
• かけるN3「かける」 also attaches to the ます form of a verb, but it does not emphasize 'starting'; instead it describes an action that is partway done and incomplete (「読みかけ」 is something read only halfway), or something that almost happened. 「始める」 looks only at the point of starting, so the two cannot be interchanged.
• ようになるN4「始める」 states only the point in time when an action or change begins, while 「ようになる」 — which attaches to the dictionary form, potential form, or ない form — expresses an ability or habit gradually taking shape. To say that someone has 'come to be able to speak', use 「話せるようになった」; 「始める」 cannot be used here.

始める neutrally marks a beginning, while 出す stresses a sudden or unexpected start. 続ける (continue) and 終わる (finish) mark later stages of the same action, かける means something is only partly done or about to happen, and ようになる expresses a change in ability or habit ('come to do') rather than the start of an action.