tabehajimeru (食べ始める):

This is a compound verb made up of "taberu" (to eat) and "hajimeru" (to begin).

It means "to start eating" or "to begin eating."

Example: 彼は昼食を食べ始めた。(Kare wa chūshoku o tabehajimeta.) - "He started eating lunch."

tabetehajimete (食べて初めて):

This phrase combines the te-form of "taberu" (to eat) and "hajimete" (for the first time, only after).

It means "only after eating" or "only when you eat."

It implies that something becomes clear or possible only after the action of eating.

Example: その料理を食べて初めて、そのおいしさがわかる。(Sono ryōri o tabete hajimete, sono oishisa ga wakaru.) - "Only after eating that dish do you understand its deliciousness."

Example

彼女と別れて初めて、彼女の大切さに気づいた。

(Kanojo to wakarete hajimete, kanojo no taisetsusa ni kizuita.)