Besides the radical, what meaning does the rest of a Chinese character have (especially when it was originally created)? I am a beginning student of Mandarin Chinese. There is a key thing about Chinese characters that I am trying to understand. Besides the radical, what purpose does the rest of a character have? Was it put there originally (long ago in history) only to help with pronunciation?
For example, &quot;begin/start&quot; - 开始. I am mainly concerned with character 始
The character 始 has the radical of &quot;female&quot;: 女
The rest of character 始 is composed of two other characters. When character 始 was originally created (as far as we know), were the two other characters only put there to help people pronounce the overall word 始, to provide additional meaning, or both?
In order words, was the radical the only part that was originally supposed to carry &quot;meaning&quot;?
If that is true, then why would there ever by more than one additional character to indicate pronunciation? It seems one character would always be enough to provide help with pronunciation.
Thank you for any insight into this.
Radicals do not have anything to do with meaning.
I'll start off with a rather lengthy correction to a common misconception. Radicals (部首, literally section 部 header 首) are just a dictionary organisation tool, specifically used for texts which collate and refer to a large collection of characters, serving as the head of each section of a dictionary. They are functionally equivalent to first letters of English words, as English dictionaries organise words under the first letter of each word.

When characters were originally created, there was no such thing as dictionaries, and certainly then there was no such thing as radicals. Since dictionaries must categorise all characters in use under a dictionary section header, the radicals of a significant number of characters were arbitrarily forced on to them, and really have nothing to do with the character's functionality. For example, see Radical 3 「⼂」, with meaning terminal punctuation mark:

Radical meaning
《說文》：「『丶』，有所絶止，『丶』而識之也」
&quot;「⼂」, if there is a place of termination, 「⼂」 is that which marks this place.&quot;
Radical sound
「主」 (Mandarin: zhǔ)
Characters grouped under this radical:

「丷」, handwritten form of two strokes which may appear as 「八」 in print shapes
「丸」, wán, small round object
「丹」, dān, cinnabar (the colour)


This is not to deny that a large number of characters do have radicals which appear to hint at the character's meaning, but this is far from a rule. 「始」 is one such character, but using proper terminology, we do not use the word radical to describe 「女」 in 「始」; rather, this meaning hint is called a semantic component.
This terminology is not just a matter of being pedantic, as many (semantic) components are not radicals at all. For example,

The character 「哥」 (to sing, now complexified into 「歌」) is made up from two simultaneously semantic and phonetic 「可」 (to sing). 「可」 is not a radical.
This is an example of complexification by piling on more components due to overuse/phonetic loaning: To sing was originally written as 「可」, but since this character became phonetically borrowed for the meaning can, able to, it was duplicated to form 「哥」, which later became borrowed for the meaning elder brother, so the meaning to sing was further complexified into 「歌」.


Now, for the rest of the query:
The rest of character 始 is composed of two other characters.
This is not correct: the rest of the character is only comprised of one other character. The character 「始」 is comprised of 「女」 and 「台」, not 「女」 and 「厶」 and 「口」 (you've performed recursive decomposition here, which generally is a very bad idea!). Don't worry about not getting this right, as this is not something you are expected to know intuitively - instead, you should rely on an accurate source of character breakdowns to give you this information. Bad reference material will not break down characters properly, and give you incorrect information.

「始」 (Baxter-Sagart OC: /*l̥əʔ/, elder sister, i.e. the &quot;first female sibling&quot; &gt; initial, first) is comprised from semantic 「女」 (female) and phonetic 「台」 (/*lə/).

... only put there to help people pronounce the overall word 始, to provide additional meaning, or both?
In general, characters have an arbitrary number of components, and each one of these components could be for meaning, sound, or both. It's not up to you as a learner to figure out; it's up to a good reference material to tell you.
then why would there ever by more than one additional character to indicate pronunciation? It seems one character would always be enough to provide help with pronunciation.
Because history is long, and characters may change over time in ways which reflect the contemporary language or change as a result of accumulating scribal errors. For example,

「耻」 (/*n̥rəʔ/, humiliation/shame) is made up from phonetic 「耳」 (/*C.nəʔ/) and phonetic 「止」 (/*təʔ/). The 「止」 part was from a series of shape corruptions from the original semantic component 「心」(heart, indicating emotions). 
See What is the history of the character “耻”? for a more detailed treatment of the glyph origins.