If you're learning radicals, Jisho.org's "kanji builder" is a pretty good practice ground.
Pick an arbitrarily complex kanji and looking at the radical table on Jisho, try to compose it by clicking on the radicals.
Keep doing it until you can't recognize any more radicals. The fewer the kanji meeting the criteria are shown, the better you are. Though be aware that the same radicals can make different kanji, so it's ok to have more than one solution.
About radicals having a meaning, >>15228
is right for the most part. Kanji originated from China and in Japan they were used more for the associated sound than for the actual meaning. Of course basic concepts like those connected with "people" might share the same radicals, but you should consider that more of an exception than a rule.
猫 being built from rice field, hands and dog is just because when the Japanese adopted that kanji, in China it was pronounced "neko" (or it was mispronounced as "neko" by the Japanese.) Nowadays it might even be that the Chinese don't even use that kanji any more (I don't actually know Chinese.)
Consider also that in modern Japanese, some kanji are misused for visual effects, like how 野 is used in stead of の to make it look like something from the ancient times.