I have no specific knowledge of Africa, mostly asia, but here are the general principles.
You can connect to internet via cable, just as the stationary computer you are using now.
You can also connect via wireless connection also called WiFi or W-lan. You then need a laptop with this function build in or a card (PCMCIA or short:PC card) to install in a slot in the laptop.
You will normally find possibility to use these connections in the more expensive hotels but also in a few internet cafés.
But your best bet to get online will be, as described, to prepare your correspondance on the laptop, transfer it to a USB memory stick and insert this in a computer in an internet café and copy/paste it to your mails, as inet cafés are getting abundant in most countries and rather cheap when out of Europe.
Funny thing is, that they are not abundant in Europe, I guess because people don't need them, as everybody has a computer in their homes.
A typical mail without HTML or attachments will be 2-3 kb thats 3-500 mails pr kb, but if your webmail provider has a lot of ads on his page it can be a lot more.
You can not rely on roaming services( your inet provider has to have agreements with the providers in each state you pass) to work always.
To be sure, to have contact, where ever you please, you'll have to hire a satelite phone with data capabilities
There is also a (old) way of connecting: via modem. I f.x. Egypt, there is a telephone number for free inet access, its slow but works ok when only transferring mails. China also has free inet access via telephone modem.
I have also used my GSM mobile as a modem. If you can't get a connection in the country you are in, you can call another country, that has roaming with your provider, you could even call the provider directly in your home country(a bit expensive, but if its urgent) if they still have a telephone conecction to the internet.
May you enjoy peace and good health !