The Return of 'As I German Learn', and A Hearty 'Fuck You' to CBS

So CBS has shunted Harper's Island to the television graveyard of Saturday night. Who didn't see this coming? I did, but I was an idiot and watched anyway and stupidly got invested. Not in the characters, mind; they're so cardboard, they're breathing Frosted Mini Wheats with sex organs, but in the mystery. I liked playing armchair detective and guessing whodunnit, and now it's doubtful that I'll ever find out if I was right, because knowing network TV, Saturday airings will soon give way to webcast airings, and my creaking systems aren't equipped to handle the technology.

I knew I shouldn't've bothered tuning in. Screw you, CBS.

I tinkered about with my German books yesterday, and lo, I have questions concerning prefixes. I know that prefixes in German can be either separable or fixed and change the meaning of the verb to which they are attached, but I'm desperately searching for the pattern by which they change them. Right now, no prefix seems to alter the verb in the same manner, and none can be attached to a specific meaning.

Kommen=to come

ankommen=to arrive

bekommen=to receive

and

suchen=to look for

besuchen=to visit

In most cases, the new verb bears little relation to the stem. Sure, "kommen" and "ankommen" are similar, I guess, at least to a brain acculturated to inexact, lazy English, but "bekommen" doesn't pertain to people or time at all. Because there is no apparent rhyme or reason to the effects of a prefix on a stem, I can't rely on inferential context to determine meaning. So, when the text cheerfully prompts me to fill in the correct prefix in order to create a coherent sentence, I look like an owl-eyed drunk trying to bluster my way through a field sobriety test. For all I know, I've just made the "brave German soldiers" fuck the dog and braise the widow in wine sauce.

Is there a systemic way in which prefixes alter verb meaning, or is this simply a matter of memorizing metric asstons of vocabulary?