[SuperZen]

Можно через reduce попробовать )

<script>
	var initialArray = ['apple', 'banana', 'orange', 'avocado', 'cheese', 2];
	var searchingString = 'a';
	var result = initialArray.reduce((accumulator, currentValue, currentIndex, array) => {
		if (typeof currentValue === 'string' && currentValue.split('')[0] === searchingString) return [...accumulator, currentValue];
		else return [...accumulator];
	}, []);
	console.log(result);
</script>

<script>
	var initialArray = ['apple', 'banana', 'orange', 'avocado', 'cheese', 2];
	var searchingString = 'a';
	var result = initialArray.reduce((accumulator, currentValue, currentIndex, array) => {
		if (typeof currentValue === 'string' && currentValue.charAt(0) === searchingString) return [...accumulator, currentValue];
		else return [...accumulator];
	}, []);
	console.log(result);
</script>