I have an eMachines computer, about as old as yours is. Yours is 64-bit, while mine is 32-bit. Therefore, if you go up to 4 GB of RAM, you will see improvement in performance. (My eMachines computer maxes out at 2 GB of RAM.) An SSD will also help; but I think the additional memory will help more. Actually, they help in different ways; so if you go with both the memory and the SSD, you will see the best improvement.
Get a new Ethernet card, if you're going to use an Ethernet cable to connect it to your router. Or get a new USB wifi adapter. Either of these will likely give you a faster internet connection than you are now getting with the Ethernet adapter which came with the computer; but I would go with the Ethernet card and Ethernet cable over the wifi adapter, if you can easily run a cable from the computer to the router.
Get a fast browser, such as Opera with ad blocking turned on. Or, to have even faster browsing, use Firefox with the NoScript security suite browser add-on. Going with NoScript means that it will take you some time to do fine-tuning. In other words, the reason NoScript speeds things up is because it blocks the incessant scripts that so many websites run in the background. But if you block ALL scripts, many sites won't run, or they will run only partially. So you have to block all scripts, and then selectively allow only those scripts which are necessary. If you will patiently go through this process, you will find that your browsing speed has greatly improved from using FireFox with NoScript.
Opera with ad blocking turned on is almost as fast as Firefox with NoScript, and there is no fine-tuning to do.
Finally, if you're open to trying something new, I suggest that you replace Windows with Linux Mint 32-bit xfce edition. I think you will find it to be a satisfactory environment. And there is lots of free software available for Linux Mint.