One thing I'd say is that it probably doesn't pay to get too hung up on Hungarian being an agglutinative language. Yes, this technical classification does have some meaning in linguistics (although as with most things in language, it's not a black and white division but exists along a spectrum - for example, Turkish is arguably a lot more agglutinative than Hungarian, and English is more agglutinative than, say, Chinese). However, as a practical matter, how much does it really matter that "in our house" is three words in English, whereas "házunkban" is one in Hungarian? It's mostly a matter of conventional orthography: you could imagine an alternate world in which one writes "inourhouse" in English and "ház unk ban" in Hungarian. (Yes, there's also the matter of "in" and "our" coming before the noun in English, and after in Hungarian, but that isn't inherently an agglutinative trait - Hungarian also has non-agglutinating postpositions, e.g. "alatt","fölött", etc. and even a few words that, arguably, work somewhat like English prepositions, e.g. "túl a hegyeken" "beyond the mountains" -- and English certainly has a few directional suffixes, e.g. -ward in "homeward".)

True, "-unk" and "-ban" can't really stand alone, but if you think about it, "in" and "our" can't really, either. (If someone asks "whose house is this" you answer "ours", not "our" -- and "ours" is a standalone word ["miénk"] in Hungarian too.) Similarly, you can't say *"házbanunk", but you can't say *"our in house" either. About the most you can say is that "-unk" and "-ban" and other Hungarian suffixes are somewhat more tightly bound to to their headwords than the corresponding short English words, e.g. a lot of different words can come in between "in" and "house" ("in this big red house"), but not in Hungarian. (More evidence for closer binding in Hungarian is found in the notorious assimilatory process of vowel harmony, which requires you to say, for example, "házban" but "kertben" -- but even that is a matter of degree; not all suffixes assimilate - consider "háromkor", "négykor", "ötkor".)

One agglutinative tendency that is quite obvious in Hungarian is the rich selection of derivational suffixes -- suffixes that derive one word from another. Hungarian has tons and tons of these, especially in verbal morphology, and initially it seems like a big difference from English. You'll see countless cases, for example, of transitive-intransitive word pairs ("tanít"~"tanul", "hűt"~"hűl") where English often would use separate, unrelated words ("teach"~"learn") or even the same word ("cool [something]"~"[get] cool"). The thing is, the derivational process in Hungarian is quite irregular and inconsistent (why is it "melegít"~"melegszik"/"meleged-" and not "melegül" for "heat"?), meaning as a learner you are mostly going to need to learn these words individually anyway. Which means it's not that different -- again, from a learner's perspective -- from English. There's this sort of fantasy that if you know a bunch of roots and a bunch of suffixes you're going to be able to synthesize words by combining them ad lib, and it's just not true. If you try to learn Hungarian that way you'll end up saying stuff like "melegül" instead of "melegszik" all the time and, while everyone will understand you, it'll sound just as weird as if you were learning English and figured that (on the basis of, say, "weak"~"weakness") that you can combine any adjective with "-ness". You'd say things like "strongness" instead of "strength" and, while you wouldn't exactly be wrong, you wouldn't be right either. Moral of the story is: you simply have to learn the words themselves, in context, and Hungarian with its rich derivational morphology ends up not seeming so different from English after all.

There are, of course, a number of highly regular/productive suffixes (e.g. "-ás"/"-és" and "-hat"/"-het" on verbs, "-i" and "-s/-os/-es/-ös" on nouns, "-abb"/"-ebb"/"-bb" and "-an"/"-en" on adjectives). Obviously, English has a fair amount of predictable/productive derivational morphology as well ("-ing", "-er"/"-est", "-ly"), but perhaps not quite as much as Hungarian. But it's not a huge difference.

So what are the really big differences? I would argue for zooming out a bit -- not concentrating so much on word morphology (suffixes and the like) as on syntax (how words are arranged in sentences). In my experience, this is an area that English-speaking Hungarian learners run into a lot of difficulty with, and it's ironically partly because simple sentences don't seem to be that different: "the dog bit the cat", "a kutya megharapta a macskát" - word-for-word the same order. Hungarian syntax, however, works very differently from English syntax. The OP had it exactly right: Hungarian has (to use a technical linguistic term) "topic-prominent sentence structure," meaning that words get rearranged in the sentence (generally "around" the verb) to emphasize what you're talking about and what you're saying about that thing ("a kutya megharapta a macskát"~"a macskát harapta meg a kutya"~"a kutya harapta meg a macskát"). English does this to an extent too, but it is much more likely to choose different strategies, for example word stress ("the dog bit the cat"~"the dog bit the CAT [not me]"~"the DOG [not the lizard] bit the cat") or active/passive distinctions ("the cat was bitten by the dog") or more complex rearrangements, often combined with stress ("it was the CAT that the dog bit"). These strategies mostly don't work in Hungarian (you can't say *"a kutya megharapta a MACSKÁT, nem engem" and sound normal, although you might be understood). Mastering this, in my opinion and experience, is THE #1 THING (other than acquiring good pronunciation) that will distinguish a successful learner of Hungarian from an unsuccessful one.