Usually the IP address you use to connect to the internet through your modem is dynamically assigned by your ISP when the modem logs on. If you restart your modem, it will be given an IP from a pool of spares that the ISP has, and while it might be the same as the old one, it's unlikely. To find out what your public IP address is, use
www.whatismyip.com.
You can specifically prevent remote access to your computer by right-clicking the My Computer icon on the desktop (or in Windows Explorer) and choose Properties. On the Remote tab, make sure the two boxes are unchecked.
This should be the best and easiest precautions. You can and probably should do a full scan for spyware. This stuff should prevent what you're worried about in 99.999% of cases. If they still get access after this, they should be working as a professional hacker/cracker making loads of money and evading the law rather than repairing/upgrading home computers!