To get connected at home the minimum you will need is:

An Internet Compatible Computer
Most recent computers have this capability.

A Modem

Computers which don't already have an Internal Modem can be connected with an External Modem with one cable plugged into your computer and another into a telephone socket.

A Telephone Line

An Internet Service Provider (ISP)
An ISP is a company which provides you with access to the Internet. This can easily be arranged through a telephone company, such as BT, or you can choose from many others. You can normally choose to pay for the duration of your connection to the Internet, or opt for a monthly payment which gives you unlimited access at any time. So remember to shop around for the method which suits you best.

Internet Software
The special Browser and E-mail software which allows your computer to connect to the internet should already be on your computer. Most Internet Service Providers also include this free of charge with your subscription.
Note: It's best to keep this software up-to-date to enable you to have the best browsing experience and to minimise security risks.