I am at loss here. So "तुम दिल्ली जाऊँगा" or "जाओगी" is wrong? How does "जाओगे" differ from the other two?
जाऊँगा is used only when you are talking in first-person (eg. मैं दिल्ली जाऊँगा). As the other commenter said, जाओगी should be accepted since it is the female form.
तुम दिल्ली जाओगे - male, singular.
तुम सब/लोग दिल्ली जाओगे - male, plural
तुम दिल्ली जाओगी - female, singular
तुम सब/लोग दिल्ली जाओगी - female, plural.
Are जाओगी and जायेगी the same?
You use जाओगी/जाओगे with तुम while जायेगी/जायेगा (also written जाएगी/जाएगा) is used with तू or with the third person singular.
Is this an imperative sentence, or is there another conjugation for future imperative?
It is not an imperative but in spoken form, the tone can be used to convey that the sentence is a command (like in English).
The closest Hindi comes to a future infinitive is to just use the imperative form of the verb. Eg: तुम दिल्ली जाना।