The Internet has changed our lives in many ways. People say it has made life better. This is probably true. It has had some bad influence, too but I think it has done more good than harm.

First of all, it has made communication much easier and it has brought around the world much closer. I have a friend in Mexico, who I HAVE WRITTEN (or WROTE????) to for years. First, I used to write her letters and I HAD to wait for weeks before I RECEIVED an answer. It took ages. Now we communicate by email. This week I HAVE SENT her five emails and I RECEIVED answers to all of them in just a few minutes.

This week I HAVE SENT - да. период времени не закончен.
I RECEIVED - да. указано, когда именно совершилось действие.

This week SO FAR I HAVE SENT her five emails and RECEIVED answers to all of them in just a few minutes.
or
I already HAVE SENT
то есть показано, что процесс еще не закончился.

I have a friend in Mexico, who I HAVE WRITTEN (or WROTE????) to for years.
здесь whom должно быть, и предлог лучше перед ним (to whom)
еще лучше другой глагол, например with whom i have been corresponding for years.

First, I used to write her letters , and ~each time~ I HAD to wait for weeks before I would RECEIVE an answer.

It took ages. - это вообще не в тему. вы только что сказали it took weeks

Now we communicate by email.

This week I HAVE SENT her five emails and I RECEIVED an answer to ~each~ of them in just a few minutes. - на каждое вы получили быстрый ответ, не на все сразу, да?