Interesting subject. My German is poor but I have spent a little time in German speaking countries.
High German is the standard - every German speaker can at least understand it and nearly all can speak it. It's what you learn if you go to German classes and it's what you're expected to speak as a foreigner. As a German learner, it's interesting and sometimes funny to notice the differences in the local dialect but generally the dialects are just a curiosity for language learners. In the worst case (if your German is good enough or you are a native speaker) where people are particularly sensitive, your attempt at the dialect could be taken the wrong way and be seen as insulting - like a foreigner talking "Paddy" to you. More often than not, when your German is rubbish like mine, most seem to find it outrageously funny when you accidentally pronounce a word using the dialect.
The analogy with heavy accents in English does not explain the situation fully in dialectised German speaking parts of Europe where the speakers can switch back and forth between the high form of the language and their local dialect. Obviously there can be also heavy regional accents but that is a different feature of the language to the high/low split. I'm not trying to be funny but you'd never say something like "I can understand Kerry but I wouldn't attempt to speak it to a native - everyone there learns to speak English well in school anyway and they'll switch to it if they know you're not local" while it would be would be perfectly reasonable to say something like that about the relationship between Swiss and high German. In some cases, children have to be explicitly taught the high form - with a different vocabulary and grammar - having already learned the low form at home and with their friends.
By the way, this high/low split isn't just a German thing - it happens frequently in other languages (look up diglossia on wikipedia) but is uncommon in English. It's how French, Italian, Spanish and others formed - all were originally local dialects of (what was called "vulgar") Latin while religious, legal, philosophical and diplomatic communications all occurred in classical Latin.
German has the reputation of being difficult to learn but as a result I think you get the feeling that this makes your poor efforts at it even more appreciated by native speakers. Also, let's face it, German isn't exactly the most fashionable language you could take on. Italian, French and Spanish are considered far sexier.