Надеюсь, мы увидимся в выходные.

Yes. Надеюсь is the 1st person singular form of the verb надеяться (to hope), so the subject я is already built into the verb ending.

• Надеюсь = I hope
• Я надеюсь = I hope too, but with the pronoun stated explicitly

Russian often omits subject pronouns when they are clear from the verb form. So Надеюсь, мы увидимся в выходные is completely natural.

The comma is standard here because Надеюсь is functioning a bit like an introductory phrase, similar to I hope at the beginning of an English sentence.

So Russian writes:

• Надеюсь, мы увидимся в выходные.

This punctuation helps separate the speaker’s attitude (I hope) from the main statement (we’ll see each other on the weekend).

You may also see other similar sentence-openers set off by commas, such as:

• Думаю, ... = I think, ...
• Кажется, ... = It seems, ...

Увидимся comes from the verb увидеться, which means to see each other or to meet.

The ending -ся often gives a verb a reflexive or reciprocal meaning. Here it is reciprocal: the action happens mutually between two or more people.

So:

• увидеть = to see
• увидеться = to see each other / meet

And:

• мы увидимся = we will see each other

So this is not just we will see something; it means we will see one another.

Because Russian uses the perfective verb here: увидеться. Perfective verbs typically express a completed single event, and in the non-past form they usually refer to the future.

So:

• мы увидимся = we will see each other / we’ll meet

This fits the meaning of hoping for a meeting later.

If you used an imperfective verb instead, the meaning or style would be different and less natural in this sentence.

Here в выходные uses the accusative case.

The phrase means during the weekend / on the weekend, and with expressions of time, в often takes the accusative:

• в понедельник = on Monday
• в среду = on Wednesday
• в выходные = on the weekend / over the weekend

The word выходные looks the same as nominative plural here, but in this expression it is understood as accusative plural.

Because in Russian выходные literally means something like days off or weekend days, so it is normally used in the plural when talking about the weekend.

• выходной can mean a day off or function as an adjective
• выходные = the weekend / days off

So в выходные is the normal way to say on the weekend.

Both are common, and both can mean on the weekend / over the weekend.

• в выходные — very common and neutral
• на выходных — also very common in modern speech

A learner should recognize both. In many situations they are interchangeable, though speakers may have personal or regional preferences.

So:

• Надеюсь, мы увидимся в выходные.
• Надеюсь, мы увидимся на выходных.

Both are natural.

Yes. Russian word order is flexible, although the original sentence is very natural.

Possible variations include:

• Надеюсь, мы увидимся в выходные.
• Мы, надеюсь, увидимся в выходные.
• В выходные, надеюсь, мы увидимся.

These all keep roughly the same core meaning, but the emphasis changes slightly.

The original version is the most straightforward and neutral for a learner.

A careful pronunciation guide is:

• Наде́юсь, мы уви́димся в выходны́е.

Approximate stress:

• на-ДЕ-yus'
• mɨ
• u-VEE-dim-sya
• v
• vy-ha-d-NY-ye

A few useful notes:

• ё in Надеюсь is always stressed: Наде́юсь
• увидимся has stress on ви́
• выходные has stress on the last syllable: -ны́е

Yes. Увидимся! by itself can mean something like:

• See you!
• See you later!

It is a very common conversational expression.

So in your sentence, the full meaning is:

• Надеюсь, мы увидимся в выходные. = I hope we’ll see each other on the weekend.

And on its own:

• Увидимся! = See you!