How to truncate a string to a specific length if it is longer?

Another way to do this, especially addressing the specific question,

My goal is to make it to 120 characters, retaining the left side

Use the left(length) method from the String class

and here's a usage example:

String s1 = 'abcdaacd';
String s2 = 
s1.left(3);
System.assertEquals('abc', s2);

and for your example would be,

string sizeString = 'Imagine this is more than 120 Characters';
string resizedString = '';
if(sizeString.Length() > 120){
    resizedString = sizeString.left(120);
}

If you don't want an ellipsis character, use a quick regular expression to match only the first 80 (or whatever number) characters, and replace the string with just those.

yourString.replaceFirst('^(.{80}).*', '$1')

As an indicator to the user that the string has been shortened, also adding '...' to the end can be helpful.

There is a String method that does that:

String newString = sizeString.abbreviate(120);

This example from the help illustrates how the length of the '...' is also considered:

String s = 'Hello Maximillian';
String s2 = s.abbreviate(8);
System.assertEquals('Hello...', s2);
System.assertEquals(8, s2.length());

Use substring method of String class

Returns a new String that begins with the character at the specified zero-based startIndex and extends to the character at endIndex - 1.

String sizeString = 'Let's Imagine this is more than 120 Characters';
Integer maxSize = 120;
if(sizeString.length() > maxSize ){
    sizeString = sizeString.substring(0, maxSize);
}