I was banned by my computer's ip address from a particular website and I wanted to get around that by changing my computer's ip address. My question is will this mess up my internet connection or do anything else unforseen to my computer?
You can try to refresh your external IP address (the one your ISP provides and the one being tracked/banned), but that isn't likely to happen. Typically, ISPs assign the same IP address to the same MAC address (the hardware address on your router).
Another way to change would be to switch to a different service provider.
So, why were you banned? The website in question must have had a reason to un-invite from the party (so to speak).