From your nuance in English...  A direct translation would not give the same thing.

我的夏天

I want to tell about my summer vacation and it would be great to receive some correction/feedback. Translation:

我要告诉和你们共享我的夏天。[Sounds demanding in Chinese. The nuance is not the same as in English. This translates to "I would like to share my summer with you all.]

这个夏天是非常愉好 [again nuance, "very well implies joy in this context, but in Chinese needs some extra oomph]。但是还要做一点打工了一下。 这是好的，因为我没有太多的工作。 沒关係, 沒有太多要做
在8月时，我不太很高兴，因为我的手机坏了。不过買但是我受到了一个新的。 

我和我的女友搭旅行了火车前往去维也纳。在那里，我们参观这了漂亮的城市。
我们也有到/去商场买了一些点儿东西。在餐厅里我们有吃了鱼。在旅馆里我们有看了世界足球锦标赛, 見到我们看了德国反对美国的击鬥。 一个周后，归我们又回到了林茨。

在家里，我们可以再一次享受我花园的美景。我常常去了游泳，是因为夏天太很热了。但是这也时常下雨了很多. 我不喜欢。
我也做了： 我常常骑了在马背上。我和我的朋友们看了一场电影。不时我学温习了经济学和 中文汉语。

I want to tell you about my summer.
This summer was very well. but I had to work a bit. That's ok, because I didn't have too much work.
In August, I was not very happy because my phone was broken. But However, I received a new one.
Me and My girlfriend and I traveled by train to Vienna by train. There we visited the beautiful city.
We went to the mall to do some shopping. In the restaurant, we ate had fish. In the hotel, we saw the World Soccer championship. We saw Germany against the United States. A week later, we went back to Linz.
At home we could enjoy the garden again. I often went swimming, because the summer was hot. But Still it also rained a lot, which I didn't like.

I also did: I often rode on go horseback riding. Me and My friends and watched a movie. From time to time I studied economy economics and Chinese.

我的夏天

I want to tell about my summer vacation and it would be great to receive some correction/feedback. Translation:

我要告诉你们我的夏天。

这个夏天是非常好。(是 should be omitted here)

但是打工了一下。

但是我不得不去工作一阵。

不得不=have to/has to/ had to

一阵= a few days/a couple of days, etc.

这是好的，因为我没有太多的工作。

这没什么，因为我的工作量不大。

If you want to say that it is just okay, just fine, 没什么 is enough and means "not a big deal".

"not have too much work"=not a large work volume=不大的工作量/工作量不大

在8月，我不很高兴，因为我的手机坏了。但是我受到了一个新的。

8月时，我不太高兴

When we say in a certain month, it will be more natural to say '“……时”;

不很高兴 is a little weird and 不太=not so/not that much/not...very

但是我收到了一个新的。

收到=receive something/hear from someone but 受到 is more used to described as like suffering or receive a punishment or being threntened and so on.

我和我的女友旅行了火车前往维也纳。在那里，我们参观了漂亮的城市。

我和我的女友搭/搭乘火车前往维也纳。or 我和我的女友搭火车去维也纳旅行。

when Chinese board a kind of transportation, the most general verb we use may be 搭(spoken)/搭乘(formal).

And 我和我的女友搭火车去维也纳旅行 is better for me because 旅行 makes your purpose heading to Vienna is much clear. 

我们去商场买一点儿东西。在餐厅我们吃了鱼。在旅馆我们看了世界足球锦标赛。我们看了德国反对美国。 一周后，我们又回到了林茨。

我们看了德国反对美国。我们看了德国对战美国的比赛。

I can hardly explain this for you. but it means "we saw the game of Germany against the United States."

在家里，我们可以再一次享受花园美景。我常常去了游泳，因为夏天很热。但是这也下雨了很多，我不喜欢。

我常常去游泳，因为夏天很热。但是这也经常下雨/这也下了很多雨，我不喜欢(这样)。

但是这也下了很多雨 probably match best with your English version.

我也做了： 我常常骑了在马背上。我和我的朋友们看了一场电影。不时我学习了经济和汉语。

我还做了：我常常骑在了马背上/我常常去骑马。
我时不时学习经济和汉语。

Chinese people like to put time/day in the middle between subjective and a verb.

For example, 我每天学习英语。I study English everyday.

If you have any question about this correction, please let me know.

我的夏天

I want to tell about my summer vacation and it would be great to receive some correction/feedback. Translation:

        我要告诉你们我的夏天。这个夏天是非常美好的。虽然打了一阵子的工，但这是好的，因为我原先没有太多的工作。
       在8月，我不是很高兴，因为我的手机坏了。但是我收到了一个新的。 我和我的女友搭火车前往维也纳，经历了一次愉快的旅行。在那里，我们参观了许多漂亮的城市。我们去商场买了一点儿东西，在餐厅里吃了，在旅馆看了世界足球锦标赛。那是一场美德之间的比赛。

        一周后，我们又回到了林茨。在家里，我们可以再一次享受花园美景。我常常去游泳，因为夏天很热。但是夏天也会下很多雨，我不喜欢。
        我还做了一些有趣的事:我经常去骑马;我和我的朋友们看了一场电影;我还学习了经济和汉语。

I want to tell you about my summer.
This summer was very well. But I had to work a bit. That's ok, because I didn't have too much work.
In August, I was not very happy because my phone was broken. But I received a new one.
Me and my girlfriend traveled by train to Vienna. There we visited the beautiful city.
We went to the mall to do some shopping. In the restaurant we ate fish. In the hotel we saw the world soccer championship. We saw Germany against the United States. A week later, we went back to Linz.
At home we could enjoy the garden again. I often went swimming, because the summer was hot. But it alo rained a lot, which i didn't like.
I also did: I often rode on horseback. Me and my friends watched a movie. From time to time I studied economy and Chinese.