Lalabas ako kasama ang aso namin

Sinabi ako sa asawa ko “Lalabas ako at ang aso aming (not needed because it's already implied that the dog is yours) sa labas”.
Madalas ako pumupunta kasama ang aso sa gubat o sa dagat, parehong malapit sa
bahay natin,
Pero Ang panahon ay medyo malamig at maulan, tuloy umuuwi ako nang mabilis.
Itinanong ng aswamo (not sure if you mean "asawa" or another word here) sa akin kung bakit mabilis ako umuuwi. 
"Kasi ang panahon ay pag-ulan at gutom na gutom ako. ano ang pagkain
mamayang gabi"

I told my wive ""I go out with the dog"
Usually I walk to the forest or the sea, both near my house.
But it was wat cold and rainy today, so I returned quickly.
My wive said to me "Why are you back so soon"
"Because it rains outside and I am starving, what do we eat tonight?"

The corrections made by si_kano would be closer to how we might normally narrate your story.  My version is to keep the corrections as close to your original statements as possible.

Pupuntahan ako at ng aso aming sa labas

Pupunta ako at ang aming aso/aso namin sa labas = I and our dog will go outside.

Sinasabi (sabi) ako asawako “Pupuntahan ako at ng aso aming sa labas”.

Sinabi/Sabi ko sa asawa ko, “Pupunta ako at ang ating aso/aso natin sa labas”. = I told my wife, “I and our dog will go outside”. (Use “ating” or “natin” (inclusive) in this case because you are talking to your wife and therefore, reference to your dog would be as a shared possession).

Madalis akong at ng aso magpunta sa gabut o sa dagat, parehong malapit sa bahay natin,

Madalas ako at ang aso magpunta sa gubat o sa tabing-dagat na parehong malapit sa bahay namin. = I and the dog often go to the forest or the shore, which are both near our house. (“Tabing-dagat” = shore; “dagat” = sea; “na” = which are. Use “namin” (exclusive) here because you are now addressing the reader who is not an owner of your house.)

Pero Ang panahon ay medyo malamig at maulan, dahil ibinalik ko mabalis (umuwi ako mabalis).

Pero ang panahon ay medyo malamig at maulan at dahil dito mabilis kaming umuwi. = But the weather was somewhat cold and rainy and because of this, we went home quickly. (You may replace “at dahil dito” with “kaya” (that’s why).  I also used "kaming umuwi" to include your dog.  If it's "akong umuwi", it will just be you that went home).

Ang sinabi aswamo sa akin ”bakit umuuwi ako mabalis?”

Ang tinanong/tanong ng asawa ko sa akin, “bakit ka umuwi agad?” = The question of my wife to me (My wife asked me), “why did you come back so soon?” (I replaced “sinabi/sabi” (told) with “tinanong/tanong” (asked) to keep it consistent with what your wife did, i.e., she asked you a question. "Umuuwi" = going home; "umuwi" = went home.  Also, it would be more appropriate to use “agad” (so soon, right away) here instead of “mabilis” because “mabilis” is descriptive of an action. In your previous sentence, “mabilis kaming umuwi” has the meaning of you rushing home. However, your wife did not see the manner by which you returned home. She was only aware that you were back home so soon.)

"casi ang panahon ay pag-ulan at gutom na gutom ako. ano ang pagkain mamayang gabi"

“Kasi umuulan at gutom na gutom na ako. Ano ang pagkain natin mamayang gabi?” = Because it is raining and I am already starving. What is our food for tonight?”  ("Ang panahon ay tag-ulan" = it is the rainy season; "pag-ulan" or "pag-umulan" = when it rains)

I told my wive ""I go out with the dog"

Usually I walk to the forest or the sea, both near my house.

But it was wat cold and rainy today, so I returned quickly.

My wive said to me "Why are you back so soon"

Pupuntahan ako at ng aso aming sa labas

literally, this can be corrected to

"Ako at ang aming Aso ay pupunta sa Labas" or "Pupunta ako at ang aming Aso sa Labas"

or you can simply omit the word pupunta "Ako at ang aming Aso ay Lalabas"

Pupuntahan is "going", it's in the form of noun, so you can't use it as a verb.

Lalabas is going out in english.

But base on what you want to say, i suggest an alternative title:

[Ako at ang aming Aso sa Pasyalan] ~ Me and Our Dog's Day out ; or

[Pamamasyal Ko at ng aming Aso] ~the same (Me and Our Dog's Day Out)

Pasyalan here is a place of recreation, it may be a park, or any indoor or outdoor places.

Pamamasyal means the action of going to Pasyalan or to the recreational places. 

As you can see, i capitalized the nouns, because that's what we usually do for titles. But you can skip that, let's focus on your translation.

Sinasabi (sabi) ako asawako 

Sinabi ko sa asawa ko or Sinabi ko sa aking asawa

Means i told my wife

Sinasabi - present tense ; sinabi - past tense 

asawa ko are two words, it is "my wife" [my - akin, ko] [wife - asawa]

“Pupuntahan ako at ng aso aming sa labas”.

Ako at ang ating aso ay pupunta sa labas, or Pupunta ako at ang ating aso sa labas, or

Lalabas ako at ang aso natin.

But just like my suggestion, you can say Mamamasyal ako at ang ating aso

Sometimes we say Ipapasyal ko lang ang aso "I'll take the dog/walk the dog out"

Madalis akong at ng aso magpunta sa gabut o sa dagat,

Madalas kaming magpunta ng aking aso sa gubat o sa dagat, or

Ako at ang aking aso ay madalas magpunta sa gubat o sa dagat,

parehong malapit sa bahay natin,

parehong malapit sa bahay namin.

namin - our

Pero Ang panahon ay medyo malamig at maulan,

~i think this sentence will do

dahil ibinalik ko mabalis (umuwi ako mabalis). 

kaya mabilis akong bumalik (ng bahay.)

dahil is because in english.

ibinalik as returned is use for things like returned book "ibinalik na libro"

Ang sinabi aswamo sa akin ”bakit umuuwi ako mabalis?”

At sinabi ng asawa ko sa akin "bakit bumalik ka agad?"

'At' here states the next action happened, it can be omitted.

'asawa mo' is your wife in english, and that is two words also.

'agad' here is soon in english. 

"casi ang panahon ay pag-ulan at gutom na gutom ako. ano ang pagkain
mamayang gabi"

Umuulan na kasi, at gutom na gutom na ako. Ano ang hapunan natin or anong pagkain natin ngayong gabi or mamayang gabi?

Umuulan - raining in english.

In your sentence, when you say the current weather like ulan you don't necessarily need to say panahon. 

Hapunan is Dinner in english.

I told my wive ""I go out with the dog"
Usually I walk to the forest or the sea, both near my house.
But it was wat cold and rainy today, so I returned quickly.
My wive said to me "Why are you back so soon"
"Because it rains outside and I am starving, what do we eat tonight?"