Quick answer
Уви́димся!
uvídimsya — a reflexive verb: “we will see each other”
The options
Уви́димся!See you!
До встре́чи!Until we meet! — warm, slightly formal
До за́втра!See you tomorrow!
Ну, дава́й!Right, bye then! — very casual
Спи́шемся.We’ll text / write. — modern and common
Why уви́димся
It is уви́деть (to see) plus -ся, which here means “each other”: “we will see each other”. Same logic in встре́тимся (we’ll meet) and созвони́мся (we’ll call each other).
What you hear back
Дава́й!Bye!
Уви́димся!See you!
До ско́рого!See you soon!
Typical mistakes
Уви́жу тебя́ по́зже.
Уви́димся!
The English phrase translated word for word sounds odd — use the fixed reflexive form.
До свида́ния to a close friend
Пока́! / Уви́димся!
Not wrong, but formal enough to sound cold.
Check yourself
What does the -ся in уви́димся mean here?
Here -ся is the “each other” meaning: we will see one another.