"She goes to my father's house."
Translation:Sie geht ins Haus meines Vaters.
No, I'm afraid you can't say that. There are several possibilities:
1) Sie geht zum (or: zu dem) Haus meines Vaters. (genitive construction, Standard German)
2) Sie geht zum (or: zu dem) Haus von meinem Vater. (von + dative construction, colloquial but acceptable)
3) Sie geht zu meinem Vater sein Haus. (VERY colloquial, regional substandard, not recommended)
4) Sie geht zu meines Vaters Haus (genitive construction with reversed word order, dated, not recommended)
"Zu" (to) and "ins" (into the) are both grammatically correct here, although "into" isn't really what the English sentence says.
"Nach" means "to" or "after". However, in Standard German you only use "nach" in the sense of "to" with geographic place names (cities/countries) and points of the compass (left/right, north/south). A prominent exception is the fixed expression "nach Hause" (home[ward]) as in "Ich gehe nach Hause" (I'm going home).
"To my father's house" is neither a city/country nor a point of the compass and also not the same as just saying "home[ward]". That's why you can't use "nach" but have to use "zu" (or possibly "ins", but with a slightly different meaning).