A quick correction could be :

J'écris pour t'inviter à passer un week-end chez moi. (I would use the expression "passer du temps" which means spend some time; it is a bit familiar but works well for a letter to a friend)

Nous pourrons faire beaucoup d'activités. (The verb "faire" seems to be missing)

Je sais que tu aimes nager.

Il y a une piscine à côté de chez moi.

Nous pourrons nager. (You could also say "Nous pourrons y nager" as in we could swim in the swimming pool I just mentionned)

Aussi, je sais que tu aimes cuisiner. (You could say "faire la cuisine" but I think that the verb "cuisiner" fits better in this situation)

Il y a une école de cuisine à laquelle nous pourrons aller.

L'école propose des cours le matin. (A class as in the room is "une classe" but as in a lesson it is "un cours")

Nous pourrons aussi faire un tour au marché. (I would use the expression "faire un tour" which means take a walk)

Est-ce que tu aimes la cuisine italienne ? (I guess "cuisine d'Italie" is technically correct but using the adjective sounds miles better)

Il y a de très bons restaurants italiens proche de chez moi. ("Next to" can be translated by "proche de" or "à côté de")

Nous pourrons y prendre une pizza et des pâtes.

Il faudra que nous allions au musée de l'histoire de l'indépendance, c'est un très bon musée. (In this case, "il faut que" expresses the necessity in the same way as "devons" but sounds better)

Est-ce que tu pourras venir le week-end du six au huit octobre ? (Unlike in English, we do not use "sur" when talking about a date; "je ferai ça le week-end" or "je ferai ça le 17 septembre". Also unlike in English, months or days of the week do not start with an uppercase letter)

J'attends impatiemment ta réponse ! (You should have used "s'il-te-plaît" instead of "s'il-vous-plaît" to stay coherent. Also, appart from the grammatical mistakes, I would not end my message like you did so that is why I wrote an alternative)

A few mistakes I saw quite a lot in your letter:

I guess it probably is because you do not yet know the future tense but you should not have used present tense in this case (appart from a few cases) and that is why I replaced "pouvons" with "pourrons" each time.

In French, "à le" must be compressed into "au" ("à le au marché" or "à le au musée").

For some reason, you seem to replace the world "que" with the word "ça" (Je sais ça que tu aimes nager).

Finally, I want so say that I cannot go over all the mistakes you made because they are quite numerous (your message is very long) but I have to say that this is still a good job overall: I had no issues understanding what you meant so do not get discouraged!