Sabado

- "Nung" or more formally "noong" refers to an event that happened in the past which is better fitting instead of "sa" which is more of a preposition used for future events.

- "binisita ko" translates to "I visited" which, I'm assuming isn't exactly what you are trying to say. Instead, "pumunta ako sa" fits better which translates to "I went to". Apart from referring to future events, "sa" is also used as a preposition that's rougly equal to "to"; whereas "ng" is understood but very awkward.

- similarly, using "ng" would be understood but again would be considered awkward and almost confusing when translated. "ay" can be used here to mean was/is (Ay can be used as a "to be" verb that works in all tenses)

Stellar job tho champ. Tagalog has very weird conjugations for tenses and you used them all right. Keep it up! :)

It would not be correct to say, “Kwento ko sa Sabado”.  The least amount of modification we can make to correct that phrase would be, “ikukwento ko sa Sabado”, but that would mean, “I will tell the story/spill the beans on Saturday”.   I don’t think that’s what you wanted to say.

“Ang kwento ko nung/noong Sabado” would mean, “the story I narrated last Saturday”.  “My story about last Saturday” would be “Ang kwento ko tungkol nung/noong Sabado”.  “The things I did last Saturday” would be “Ang mga ginawa ko nung/noong Sabado”. 

The corrections given earlier are correct.  However, aside from the replacement of the “ng” with “ay”, as Gian Carlo suggested, I would add a “noon” (then) in your last sentence – “Pagkatapos, pumunta kami sa parke at ang panahon noon ay maulan at maulap” = Then we went to the park and the weather then was rainy and cloudy.  If what you meant was that it was cloudy and rainy also at other times that day, then I would suggest that you come up with 2 sentences instead – “Pagkatapos, pumunta kami sa parke.  Ang panahon noon ay maulan at maulap.”  Notice also that I added a comma after “pagkatapos”.  Although the sentence may be translated to “then we went...”, the back translation with and without the comma would be the difference between saying “after, we went...“ and “after we went...”.     

This is how filipinos would say it:

Pumunta ako sa sinehan nung sabado. Tapos pumunta ako sa restaurant kasama pamilya ko. Pagkatapos pumunta kami sa parke habang maulan at maulap.  

Pumunta ako sa sinehan sa nung ("noong" is the proper word but no one really says it that way) sabado. Tapos pumunta ("bumisita" is "visited" and you wouldn't say "I visited a restaurant") ako sa restaurant kasama ang pamilya ko. Pagkatapos pumunta kami sa parke habang maulan at maulap (or) nang biglang umulan ("when it rained all of a sudden".