What you could also do is change your Ip address. Your router will do this automatically whenever you turn the device (your pc) on, depending on what other connections are open and what other IP's are assigned.
Well, that's true, but the address you're thinking of is probably your internal networks IP address. Internal only networks have addresses that start with 192.168.x.y. However, your router / cable modem does translation. It has one address for your internal devices, and one address open to the world, and it pases things between the networks.
The address the OP would be worried about is NOT under his control. Its assigned by his ISP, its usually dynamic (you pay more for a static one), and typically you'll get the same one each time, unless your router / cable modem is off an extended amount of time.
The danger of someone knowing your IP also depends on your setup. If you have your PC's network card plugged directly into a cable model for example, youre way more exposed. Your computer is pretty much directly attached to the Internet. You need to be running a software firewall, and even with that its dangerous. If you have a home router, those have firewalls built it. Provided you've changed the admin password, you should be good to go. The routers include firewalls and shield all the computers plugged into it. Its possible the router has a bug someone could hack, but fairly unlikely.