I’ve always been fascinated by remote island communities, and few places are more remote and more island than St. Helena. They have a wonderful page about communications to, on, and from the island, and it’s delightful.
However you connect, the Internet on St Helena is slow and expensive! For technical details and pricing information please contact Sure.
Assuming you are a visitor you are best to access the Internet via your mobile (cell) Device. Otherwise you will not have a telephone account so will need to use one of the few Internet Kiosks, mostly in Jamestown, which are very expensive.
If you are staying longer you can sign up for an Internet access package, billed on your telephone account (this should be available even if you are renting accommodation but check with your landlord). Broadband Internet was introduced in 2007, but be aware that data transfer speeds on St Helena are considerably slower than in most other countries and monthly data transfer limits are very low.
None of the above will be surprising. There’s tons of information and history on this page, and it’s definitely worth a read.