I am not an English teacher, but I was able to find a few small errors.
You have received this letter from your pen friend.
...Thanks for inviting me to stay with you when I visit your country next month.
I'm not sure how to get to your apartment from the airport. Could you write back giving me some basic instructions? What would be the best method of transport for me? I'd prefer one that isn't too expensive!
Just one other thing - what will the weather be like when I get there? (Just so I'll know what clothes to pack!) ...
How are you
going[getting] on? I'm very happy you're coming next month. I've already told my mum that you are going to live with us some time[for a while] and she seemed to be happy. She thinks that [it] is good for me to have an English-speaking friend because I can improve my English talking to you.
If you want to, I could be waiting for you when you arrive at the airport. If not, I
only can[can only] tell you that you have to stand near some road. I'm sure that a lot of taxis are going to want to pick you up. Be careful with the money you give to the man who is driving because sometimes they try to stick with[keep] your change.
The weather will be cold because you're coming in the middle of the winter. But you don't need more than a warm jacket and a pair of gloves.
Well, I have to continue studying for my exams. I'll be waiting for your
following[return or next] letter.