To see if you are getting the bandwidth you are paying for you can search for "internet speed test" on Google OR you can type in your local ip address (usually 192.168.0.1 or 192.168.1.1) and click on "modem status" (or something similar, each modem/router interface will be different). There you can view your down/upstream rates.
If you are wirelessly connecting, make sure that your connection strength is at least 50% to be getting a decent connection.
The speed of a computer also affects the "speed" of the internet because your computer still has to compute all the data and/or render video when you view youtube or something else. Even if you have an average computer, you need to make sure you don't have a billion processes running in the background or else this will slow everything down on your computer because it is using up your RAM/processing power. Sometimes this is just from accumulation of tons of things at start up or could also be caused by a virus. You can start the task manager to view processes (push ctrl+alt+del) and see what processes are slowing down your computer.
Oh, sorry...misread your post. Given your circumstances, I would guess it is the computer itself (e.g. processes are hogging up your computing power or your computer is just sub-par in the first place)? Best of luck!