Ano ang pagkain natin mamayang sa gabi?

Ano ang pagkain natin mamayang sa gabi?
(What is our foot tonight?).
Gusto kong kumain ng gulay at tinapay.
(I'd like to eat vegetables and bread).
Wala kaming(naming) (naming is ours)gulay pera mayroon kaming(naming) maraming tinapay
(We don't have vegatables but we have plenty of bread).
alam ko, pupunta ako sa tindahan at bibili ng gulay,
(I know, I will go to the shop and buy some vegatables).
Gusto mo bumili ako ng ibang bagay pa?.
(Do you want me to buy sonthing else?).
Oo, bumili sa akin pakibili mo ako ng ilang melon pakiuasap. Salamat.
(Yes, can you buy me a melon please?).
Sige, hanggang sa muli!
(Sure, see you later).

Question: do kaming (kamin) and naming (namin) have the same meaning?

kami / kaming - we  

example.. Kami ay Filpino - We are filipino.

Meron kaming tinapay - We have bread.

namin - our

Mary is our daughter - Si Mary ay anak namin.

Ano pagkain natin mamaya sa gabi? = What is our food later, in the evening?

More correct: Anong/Ano ang pagkain natin mamayang gabi? = What is our food tonight?

Gustokong kumain ng gulai at tinapy.

Gusto kong kumain ng gulay at tinapay. = I'd like to eat vegetables and bread.

Wala kaming(naming) gulay pera mayroon kaming(naming) maraming tinapay

Wala kaming gulay pero mayroon kaming maraming tinapay. = We don't have vegetables, but we have plenty of bread. (Exclusive - does NOT include the person being told.)

Wala tayong gulay pero mayroon tayong maraming tinapay. = We don't have vegetables, but we have plenty of bread.  (Inclusive - includes the person being told.)

More natural: Wala kaming gulay pero marami kaming tinapay; Wala tayong gulay pero marami tayong tinapay.

alamko, pupunta sa tindahan at bibili ng gulay,

Alam ko.  Pupunta ako sa tindahan at bibili ng gulay.  = I know.  I will go to the store and buy some vegetables.  (Without "ako", the sentence has no subject.)

Gusto mo bumili ako ng ibang bagay pa?.

Gusto mong bumili ako ng ibang bagay pa? = Do you want me to buy something else?

More common structure: Gusto mong bumili pa ako ng ibang bagay?

More natural: "May gusto ka pang ibang ipabili?" or "May gusto ka pang ipabiling iba?"

Oo, bumili sa akin ilang milon pakiuasap. Salamat.

Oo, puwede mo akong ibili ng isang melon?  Salamat. = Yes, can you buy me a melon?  Thank you.

Oo, puwede mo akong ipakibili/pakibili ng isang melon? Salamat. = Yes, can you please buy me a melon?  Thank you. 

(Note: Adding "paki" to the root verb is equivalent to saying "please".  "Ibili" becomes "ipakibili" as the "paki" goes between the "i" and "b", "bili" being the root verb.  However, it is not uncommon to have the first "i" omitted; "sa akin" = to me; "ilang" = a few 

Sige, hangang sa muli!

Sige, hanggang sa muli!  = OK, till the next time! (Sure, see you later).

(Note: "Hanggang sa muli" may be too formal and is very rarely used, except in literature.  "Magkita tayo mamaya" would be "see you later", but is said usually when referring to a pre-arranged meeting for later - as when you and a friend agree to meet someplace after work.   In the context of your "story", "Sure, see you later" would be more like our "Sige, mamaya na lang" (OK, later).  "Na lang" is a way of saying that the speaker is referring only to "later" as a time reference and nothing else.

Question: do kaming (kamin) and naming (namin) have the same meaning?

Kami = We (exclusive)

Tayo = We (inclusive)

"Kaming" is also "kami na", i.e., "kami" when followed by a qualifier.

Ex:  Kami ay nandito na. = We are already here.

Kaming/Kami na mga manggagawa ay nandito na. = We, the workers, are already here.

(Note: There is no "kamin")

Namin = Our (exclusive)

Natin = Our (inclusive)

"Naming" is also "namin na", i.e., "nami" when followed by a qualifier.

Ex:  Ang bahay namin. = Our house.

Ang bahay naming/namin na luma. = Our old house.

Ang bahay naming/namin na malapit sa ilog. = Our house that is near the river.