"I'll make dinner" ("I will make dinner" ) is future tense, and therefore would have a different conjugation of 'hacer' - "Yo haré la cena". 'Hago' is present tense meaning "I make".
Hacer is an irregular verb. It's not going to conjugate the way many other -ER verbs do in the present tense.
Hago? I don't know this conjugation. I understand it comes from hacer though.
"Hago" is the 1st person singular, present tense form of "hacer".
Yo hago = I make/do/etc.
Tú haces = you make/...
Él/Ella/Usted hace = he/she/you(formal) make
Nosotros hacemos = we make
Vosotros hacéis = you all(formal) make
Ellos/Ellas/Ustedes hacen = they/you all make
The 1st person singular form has an irregular conjunction.
Present tense forms of "hacer":
I would guess because adding "do" is an English way of adding emphasis or improper grammar. If the latter, then it's incorrect English. If the former, then there is probably a particular way of adding emphasis in Spanish that has not been covered.
"I'm doing dinner" was wrong, even though that is a commonly used expression in English as a substitute for "making dinner". Pretty annoying given that hacer also translates as "make or do"
'Hago' (I make) is the present tense of 'hacer' (to make/do/etc.) and 'hice' (I made) is the past tense of 'hacer'.
"Hago" (1st person singular form of "hacer") means both depending on the context. "Hacer" has a variety of meanings.
The dictionary in the website Reverso is a great resource to learn the meaning of Spanish (or French, German, etc.) words, and the conjugator is a fabulous tool to learn all the different conjugations of verbs.
The link below lets you know all the meanings of and ways to use "hacer" (AKA hago).