Sometimes, of course, you do need "begin" or "start" to indicate the beginning of an action; it all depends on the context. So, check each instance carefully. Removing unnecessary uses of "start" and "begin" will make your writing more clear, and it will also reduce your word count by around 15%. Below you will find some examples where the words "start" or "begin" are unnecessary:
BEFORE: Tears started to stream down her face. (7 words)
AFTER: Tears streamed down her face. (5 words)
BEFORE: The smell of freshly baked bread made me start to feel hungry. (12 words)
BEFORE: The smell of freshly baked bread made me feel hungry. (10 words)
BEFORE: When the knight saw the queen, he began to shout with joy. (12 words)
AFTER: When the knight saw the queen, he shouted with joy. (10 words)
BEFORE: Rama quickly began to gain ground as the demon king started to grow tired. (14 words)
AFTER: Rama quickly gained ground as the demon king grew tired. (10 words)
BEFORE: Tears began to fill his eyes and started to roll down his cheeks. (13 words)
AFTER: Tears filled his eyes and rolled down his cheeks. (9 words)
BEFORE: When the salt touched its skin, the demon began to shriek in pain. (13 words)
AFTER: When the salt touched its skin, the demon shrieked in pain. (11 words)
BEFORE: We waited there, very worried, until the other search parties began to return. (13 words)
AFTER: We waited there, very worried, until the other search parties returned. (11 words)
BEFORE: After reading the diary, he began to realize why she had left him. (13 words)
AFTER: After reading the diary, he realized why she had left him. (11 words)
BEFORE: She gave her dad a hug and then began heading for the departure gate. (14 words)
AFTER: She gave her dad a hug and then headed for the departure gate. (13 words)
BEFORE: The monstrous hydra's heads started to grow back each time Hercules chopped them off. (14 words)
AFTER: The monstrous hydra's heads grew back each time Hercules chopped them off. (12 words)
For more strategies to use in reducing your word count while improving your writing, see this list: Short and Sweet Writing Strategies.
Hercules and the Hydra, by John Singer Sargent (1921)
Web Source: Wikipedia