The speed of your computer is controlled by a number of hardware factors including the CPU, memory, hard disk speed, and the speed the data moves around your computer (BUS speed).
You can replace the CPU in your computer (not easy in a laptop but fairly easy in a desktop).
You can add more memory (which is easy) and faster memory (which is probably not worth it).
You can buy faster hard disks (but probably not worth it).
You cannot do much about the BUS speed as it built into the computer (you can get a new motherboard but that is a bit drastic - again not easy in a laptop).
You can defrag your hard disk to gain a little increase in speed.
If you are running Windows then you can turn off some of the fancy user interface features to speed it up.
You can check which programs are starting when Windows starts and maybe stop some of them starting. The more programs running on your computer the slower it runs.
You could re-install Windows to get rid of all the rubbish that is gathered over the years.
Dont confuse your computer speed with broadband speed. Increasing the speed of your computer will not increase your broadband / internet speed.