First off, if you have wifi and it is not password protected, set one up. Methodology differs from router to router, but you should be able to find your router's IP address somewhere in the documentation. Type that into the address bar of your browser and login, set up password protection. If your network isn't protected, other people could be logging onto it and that will slow your connection.
You could buy a better router, that might speed things up a bit. If your internet is running slow, it could be the computer. Scan for viruses, adware and spyware, all of which will slow you down. Try a different browser, something like Firefox might get you better results than Internet Explorer, especially on older machines. Make sure there are no processes running in the background that you don't want. Some of them may be trying to access the internet, slowing down your connection.
Finally, some internet providers have an Extreme level of internet service. As someone who just switched from standard to extreme, I can tell you, you may not notice a difference in browsing, but you will when you download something.