Learn Hardcore Hungarian: Én gyalog megyek haza.

add emphasis (“It is I who am going on foot”),
or to contrast with someone else (“You drive, I walk”).
In neutral statements you’d typically say Gyalog megyek haza.

What part of speech is gyalog, and why doesn’t it look like a typical adverb?

gyalog is an adverb of manner meaning “on foot” or “by walking.” It derives from the noun gyalog (“infantry; foot soldier”), but here it tells you how you go. Hungarian adverbs often have no fixed ending and many come directly from nouns or adjectives.

Why is haza used instead of a noun with a case ending, like haza-t?

haza in this context is not the accusative hazát (“the house”), but an adverb of direction: “homeward,” “home.” Motion-to adverbs often have no extra suffix. You could also say hazafelé (“towards home”), but haza megyek or hazamegyek is more direct: “I go home.”

Can I write hazamegyek as one word instead of haza megyek?

Yes. The verb hazamegy (“go home”) plus the personal ending -ek becomes hazamegyek: “I go home.” Splitting it into haza megyek is allowed, especially when you want to insert adverbs (like gyalog) between them.

Why not use gyalogolok (“I walk”) instead of megyek with gyalog?

gyalogolok means “I walk (for exercise or stroll),” whereas megyek is “I go.” When describing your mode of travel, Hungarians prefer mode-of-transport adverb + megyek: e.g. biciklivel megyek (“I go by bike”), so gyalog megyek (“I go on foot”) is more idiomatic than gyalogolok haza.

Is the word order fixed? Could I say Haza gyalog megyek or Gyalog haza megyek?

Hungarian has flexible word order, with focus and emphasis guiding placement:

Gyalog megyek haza. (neutral)
Haza gyalog megyek. (emphasizes home as the destination)
Gyalog haza megyek. (emphasizes walking toward home)
All are grammatically correct; just watch for which element you want to highlight.