"I go home."
Domum is the accusative form of domus and domō is the ablative form. Use the accusative for motion toward (ad -> "to") and the ablative for motion from (ab -> "from"). "Go home" in English is idiomatic for "go to home", so it would require the accusative, and Latin, just like the English equivalent, does not use a preposition with domum.
DuoLingo uses an implicit rather than explicit method for teaching language, meaning that the grammar is taught through examples rather than through giving the grammar outright. The implicit method has the advantage of making you more fluent (since that's how you learned your first language), but, as you note, it does have the drawback of making it easy to not see the "big picture" grammar-wise. My personal recommendation is to use DuoLingo to get the vocab and a basic feel for the language, and then consult a Latin grammar to get a more rigorous grasp on the grammar.
The general pattern is: if a noun can take the locative case, it can also take a bare accusative (for motion towards) and a bare ablative (for motion from).
"ad urbem" -- "to the city"
"domum" -- "homeward" ([to/towards] home)
"ab urbe" -- "from the city"
"domū" (or "domō", depending on the period) -- "from home"
Interestingly, "home" is similarly special in English:
"I'm going to school"
"I'm going home"
By "period" I was contrasting earlier Latin practice and later Latin practice. Latin underwent a lot of changes from Old Latin (before 75 BC), through Classical Latin, to Late Latin (post-Classical).
"Domus" is a very messy noun, with some second declension endings and some fourth declension endings.
"Domū" would be the normal fourth declension ablative singular.
"Domō" would be the normal second declension ablative singular.
The older practice was to use the fourth declension ablative singular form ("domū"). Later on, people used the second declension ablative singular ("domō").