When I finished the Dawnguard missions, I sent Serana to be cured and she said "I will see you at home". I assumed that meant my home but I haven't found her there and it has been in-game months. Where does she go once she is cured?
According to the wiki, she will be at Dawnguard not at your home. Since she doesn't live with you it makes sense she won't be in your place. And the castle would presumably remind her of her dead father so she isn't there.
"After Serana brings her scroll to Fort Dawnguard, you ask her "Have you always been a vampire?" while she is a follower. During this conversation, if you ask her if she has ever thought about getting cured, she will refer to her condition as a gift, insisting that she will never seek a cure for it. Should she say this, you can still ask her to seek a cure after her father has been defeated, but she will refuse and then refuse to speak to you about it further should you ask again.
If you wish for her to cure her vampirism after the main questline, choose the option "Are you all right?" rather than asking her about the cure. Once you have defeated Harkon and left Volkihar Cathedral, if you are not a vampire yourself, you can ask Serana if she's ever considered being cured. If you tell her that her vampirism makes you uncomfortable, she will criticize you for being self-centered and refuse to ever discuss the cure again. If you pick more tactful dialogue options, you can convince her to go see Falion. She will immediately leave, even if she is currently following you, saying that she needs to do this alone. Within a few days, she should show up at Fort Dawnguard as a normal human."
(Bold emphasis mine)