Translation:See you next week.
In my rather limited experience, which is mostly made up of LINE conversations with my mother (a native Japanese speaker), we use また more than 又. I would definitely use the kanji if it were either in a place name, or I was speaking about parts of kanji (奴 has 女 and 又), but not in this informal greeting.
I wouldn't use Duo as a rule of thumb for whether something is more frequently written in kana or kanji since the developers and contributors are constantly working on the course and adding things. However, some dictionaries like Jisho say how a word is typically written in its entry. For example, it confirms that また is usually written in kana alone.
"See you in a week" sounds like a time span, like you will see the person exactly 7 days from now.
一週間後 - "after one week"
"See you next week" doesn't have to be exactly 7 days between seeing someone, it's just sometime within the next week.
You could see them on Friday and then again on Monday and still be correct when saying 'see you next week'
来週 - "next week, the coming week"