Hey!
From what I understood you want to connect to the internet to read e-mails.....
There are two main ways you can connect an arduino to the internet:
1.)Wi-Fi shield
2.)Ethernet shield - using an ethernet shield you can turn your arduino into a client and then read your e-mails, check out this instructable,I think it might be extremely helpful to you.
Another third way would be connecting the arduino directly to a computer......
The arduino itself has a very limited memory so downloading emails can be a challenge but most ethernet shields come with an SD card slot which can store plenty of data.
BTW what do you mean by "transmit data"?
Good Luck!