So, you’re telling me that each character has radicals (and other components), which by themselves have a meaning

Yep, for example 相 is the combination of 木 and 目. 木 means wood, and 目 means eye. You can find these radicals in a lot of characters: 校, 看, etc.

are combined with other radicals for meaning and sound, to form a character that has meaning (logographs), and then you combine characters (sometimes just for their sound) to have yet another meaning?

Not exactly. Sometimes characters and radicals are combined because they hold some kind of relationship (for example 木+木=林), and some times because the character looks like the real thing it is trying to represent (for example 鸟), but most of the time we actually don't know the exact reason why they were combined.

When combinations happen, sometimes the character takes the sound and/or some small implied meaning of its components/radicals, and sometimes it doesn't. For example, the radical 氵 in 洗 indicates that the character's meaning is related with something about water, and that is true. But in the case of the character 法, it doesn't take the implied meaning of water, nor the pronunciation of either 去 or 氵.

I mean, I get it. It’s like ‘man’ and ‘manslaughter’ but with the addition that ‘m’, ‘a’, and ‘n’ all had separate meanings.

No, radicals don't give the character its meaning, radicals are just visual cues that can help you figure out the characters meaning and/or sound (and sometimes they can't, as you can see in the example I gave you above). For example, 想 is composed by 相+心, and it's meaning has nothing to do with the meaning of its components. The meaning of the character doesn't change according to the radicals/components it contains.

I’m just wondering when Chinese will start putting in spaces between ‘words’. Probably when enough words become more than a single character, I suppose.

In Chinese, spaces aren't used most of the time. I think you are confusing words, characters and radicals.

• 家人 is a word that means "family member", it's composed by two characters.
• 家 is a word and a character, it means "home".
• 人 is a word, a character and a radical, it means "person".

• Word: can be either one character that can stand by its own and still have meaning (家), or a combination of different characters that when put next to each other form a meaning (家人)
• Character: visual representation
• Radical: common visual cue in characters

Anyway, don't think too many about radicals, they aren't really important for learning characters/words. Focus on learning characters/words.

I hope I helped, comrade.