غدا سوف تذهب أمي إلى السوق، وأنا سوف أعمل في البيت.

غدا means tomorrow, and placing it first is a common way to set the time frame for the whole sentence. It can also appear later, for example:

• سوف تذهب أمي إلى السوق غدا. Both are correct; the beginning position just emphasizes the time.

They both point to the future, but they do different jobs:

• غدا specifies when (tomorrow).
• سوف marks the verb as future (will/shall). Using both is normal and not considered redundant: one gives the time, the other gives the tense/meaning of the verb.

Both can mark the future in Modern Standard Arabic:

• سوف تذهب and ستذهب both mean will go. General tendencies (not strict rules):
• سـ is shorter and very common in writing.
• سوف can feel slightly more emphatic or more formal, but in many contexts they are interchangeable.

Because the subject أمي (my mother) is feminine, so the verb agrees:

• تذهب = she goes / she will go (3rd person feminine singular in the imperfect)
• يذهب would match a masculine subject (he goes).

أمي means my mother and is formed by attaching the possessive suffix ـي (my) to أم (mother):

• أم + ي → أمي In normal Arabic text, short vowels and some pronunciation details are not written, so you usually just see أمي without extra markings. (In fully vowelled text you might see something like أُمِّي depending on context.)

إلى is the most direct preposition for physical movement to/toward a place:

• تذهب إلى السوق = she goes to the market لِـ can sometimes mean to in the sense of for or toward, but for straightforward destination, إلى is the standard choice in MSA.

The و is the coordinating conjunction and, linking two related clauses:

• ...، وأنا سوف أعمل... = ..., and I will work... Starting with وأنا is very common to signal a new but connected clause. You could also say:
• ...، وأنا سأعمل في البيت.
• ...، سأعمل في البيت. All are possible; وأنا adds emphasis that I (in contrast to your mother) will do something else.

You often do not need it. The verb أعمل already means I work / I will work because the أ- prefix marks first person singular.
Including أنا is for emphasis or contrast:

• وأنا سوف أعمل emphasizes I (as opposed to someone else).

Arabic uses the imperfect (present-like) verb form for both present and future. The future meaning is created by particles like سوف or سـ, or by context:

• سوف أعمل = I will work
• أعمل غدا can also mean I will work tomorrow even without سوف, because غدا forces a future interpretation.

Modern Arabic writing often uses punctuation similar to English, including the Arabic comma ،. It is not “required” in a strict grammatical sense, but it is standard for clarity.
You could also separate the two clauses with a full stop:

• غدا سوف تذهب أمي إلى السوق. وأنا سوف أعمل في البيت. Both are acceptable; the comma emphasizes that the two actions are part of the same overall plan for tomorrow.

Common MSA options:

• لن
  • imperfect for future negation (strong, standard):
    غدا لن تذهب أمي إلى السوق، وأنا لن أعمل في البيت. = Tomorrow my mother will not go..., and I will not work...
• لا is usually present/habitual negation, not the standard choice for a planned future in formal MSA.
• You can also negate سوف sentences with لن (you typically do not keep سوف when using لن).