1.

"You said to me you'd travel abroad during a whole month."

To sound more informal I would suggest this: "You told me you'd travel abroad for a whole month." 

2.

I'd like to know if "really" sounds more informal than "in truth".

I would need to see the context of the sentence to answer this.

3.
"I couldn't believe my eyes when he took my number out of the box! I was feeling absolutely delighted!"

I would avoid using the word 'delighted.'  It is not wrong, but it is not commonly used in everyday speech. I would suggest something like: "I was so excited!" or "I felt so excited!" 

4.

"Besides, I think that taking this trip is a good way to improve our English."

'Besides' might not be the best choice here depending on the context and the previous sentence. Are you trying to say 'In addtion...' or 'Also..." The use of 'taking' is fine.

Depending on this I would suggest something like:

"Also, I think that taking this trip is a good way to improve my English."

Looks good!

Could someone helps me, please?

Hello! I'm trying to write an informal email in English and some doubts came to me. Firstly, I want to ask if this sentence is correct or not: "you said to me you'd travel abroad during a whole month". The issue is that I don't know if the word during is correct here, I mean, maybe the sentence is better in this way: "...abroad a whole month", without during.
In second place, I' d like to know if "really" sounds more informal than "in truth". Finally, can you tell me if I'm right saying the sentences below in this way?:

- I couldn't believe my eyes when he took my number out of the box! I was feeling absolutely delighted!
- Besides, I think that taking this trip is a good way to improve our English.

Well, the question is this. I'm not sure if in the last one "taking" is correct. For me, it doesn't sounds weird but I'm not sure... Perhaps, it doesn't fit and I just should have written " I think that take this trip", without the -ing form.
Also , in the previous one, I mean in the first sentence of this two above, I don't know if "I was feeling absolutely delighted! " is correct. I think I have a mistake there, haven't I?

I'm looking forward an answer if it's possible. I'll be very thankful if someone helps me with the questions I've asked (please give me an explanation about my mistakes, because I 'd like to understand it in order to don't have those mistakes another time) and with this entry too (I'm sure it's full of mistakes).
That's all. Sorry for my awful English.

Hope this helps:

"you said to me you'd travel abroad for a whole month"

really is more informal but depends of the context of the sentence.

Besides, I think that taking this trip is a good way to improve my English. My is used when talking about yourself. Our is used when talking about 'we' (yourself and someone else, you and your brother/sister/ friend for example)

I was feeling absolutely delighted! This fine.

I 'd like to understand it in order that I don't make these mistakes again.

Otherwise the rest is very good.