温泉

Today I intend to go to an Onsen (hot spring) with my familiy since the morning.

"since the morning" is out of place here. If you wanted to say when you came up with the plan you could say "This morning, I made plans with my family to go to an Onsen."

Although I say Onsen, it isn't the a real Onsen like Kusatu and Hakone, it is just the local bathhouse.

you could also say "Although I say Onsen, it isn't a real Onsen in the same respect as Kusatu or Hakone, it is just the local bathhouse."

I think "bathhouse" would be the best way to say in English the kind of places that have the facilities of an Onsen but in normal towns.

Do you know if there are Ganbanyoku in another countries?

( Ganabanyoku is a type of hot stone spa )

This is a better way to describe what 岩盤浴 is in English.

And There is also the a restaurant at the same place, so we can spend the day there without having to leave.

A more natural way of saying this would be "The bathhouse also has a restaurant, so we can spend the whole day there without leaving."

However, because it has been gets crowded in the afternoon, I will intend to go back home when if it has been gets too crowded.

You could also say "However, because it often gets crowded in the afternoon, I intend to leave when it gets overcrowded."

We will enjoy it, because my family will be there.

This implies that your family will be meeting there, so you will be enjoying their company, but you could also say "My family will all be meeting there, so we will surely enjoy ourselves."

Today This morning I intend plan to go an Onsen (a hot spring) with my familiy family since the morning.

今朝は家族で温泉に行く予定です。　It's more natural this way in English.

Although I say Onsen, It isn't the real Onsen like Kusatu and Hakone, it is the facilities of Onsen in the normal town.

You can say "Although I said Onsen, it's not a full-fledged hot spring like in Kusatu or Hakone, it's a facility in a normal town."

本格的 = Full-fledged. Colloquial. You will sound like a native!

There are not only Onsen, but Ganbanyoku and saunas.

"Sauna" is a word we use in English, so we pluralize it when we talk about more than one, or when we talk about saunas in general. We don't have the words Onsen and Ganbanyoku, so I think we would leave those as they are.

Do you know if there is are Ganbanyoku in another other countries?

Here, Ganbanyoku is plural.
There is a Ganbanyoku.
There are many Ganbanyoku.

Also, another = "one more in addition to" other = "different"
I want buy another hat.
I want look for other hats.

And there is the restaurant in the same facilities , we can spend our time without going out from there all day.

"It also features a cafeteria, so you can spend all your time/all day there without having to leave." <-- is how I would write that.

However, because it has been crowded in the afternoon, I will intend plan to go back home when if it has been becomes crowded.

We only capitalize the first word of the sentence and proper nouns (names of places, people, things, holidays, etc.) Also, the first part of your sentence is a little unclear. I might say "However, because it is sometimes/usually/often/always crowded in the afternoon, I plan to go back home if it gets crowded today.

We will enjoy it, because my family meets.

Are you saying that you haven't met in a long time? Sorry, my Japanese is terrible. I think you mean "It's been a while since my family met/got together, so we will all have fun."

I set off to an Onsen (hot spring) with my familiy this morning.

There are many types of Onsen.

Do you know if there is any Ganbanyoku in other countries?

There is a restaurant so we can spend our time there all day long.

However, because it would be crowded in the afternoon, I intend to go home if it really does.

I am sure we would enjoy it becuase all family members gather together.