"I am going home early today."
Not unless you want to specifically talk about what happened today. For example:
今日、レストランに行った (Kyō, resutoran ni itta) — Today, I went to the restaurant.
今日はレストランに行った (Kyō wa resutoran ni itta) — As for today, I went to the restaurant.
Both are correct, except that in the second example the speaker wants to talk about how today was. It’s up to you to choose whether you want to talk about the day or rather what you did then.
Here is a link about adverbs. You can turn many adjectives into adverbs. http://www.guidetojapanese.org/learn/complete/adverbs
"Rules for changing adjectives into adverbs
For na-adjectives: Attach 「に」 to the end. Examples: 静かに, きれいに, 本当に.
For i-adjectives: Replace the 「い」 with 「く」. Examples: 早い + く = 早く, かわいい + く = かわいく. Exceptions: 「いい」 becomes 「よく」 「かっこいい」 becomes 「かっこよく」"
Person A stands up from their desk
PERSON B: "Where are you going?"
PERSON A: "I'm going home early today."
The journey has already started, it's therefore present progressive, and I think 帰っています is an acceptable translation.
"Going" in this example could easily mean either "I am going to go," which is probably the more common usage in English, or just plain "going," which is what it actually says.
Here is an interesting perspective on 帰っています:
Also, I've heard that if someone is leaving early, they commonly say: お先に失礼します = Excuse me for leaving first.
The て-form has a few different uses: when followed by います it represents the continuous tense: "I'm going home (right now)." When followed by another verb or sentence it represents multiple actions: "I went home and opened the door." or it can be an impolite command. "Go home!"
I don't know what 「京家」is, it might be a typo, but other than that 早い needs to be 早く, since it's acting as an adverb here, and duolingo is also trying to teach us that 帰る means "to return/go (home)". There's a bit of a difference between the two, yours being that home might be your first destination, while 帰る would be like you already went to your destination, and now your returning home.
It seems that Duolingo is not translating it literally this time, "I am going home early" means that you will go home early later that day, rather than that you are already going home, and it's early. Japanese doesn't have this, as far as I'm aware, so the current translation is the only one that works.