ZoneAlarm is merely a firewall which is something that filters out incoming and outgoing connections. It may prevent an attacker from connecting to a certain program on your computer or it may prevent a virus on your computer from accessing the Internet and possibly sending out private information.
For the ultimate in spyware and virus protection I recommend this setup:
-SpyBot Search & Destroy - be sure to enable immunization and update often
-SpywareBlaster
-AdAware
-ZoneAlarm, which you already have
-Avast! antivirus - only if you don't currently have antivirus protection
-Mozilla Firefox - do not use Internet Explorer unless you absolutely have to
I believe using these programs for protection is tried and true, since I very rarely ever encounter spyware. I scan months apart and usually don't find anything.
As for the IP issue, changing it only might stop the intrusion attempts, although there are no guarantees. For instance, if you have a virus on your computer attempting to connect to China, then it doesn't matter how many times your IP address changes since the virus is revealing you.
Your IP address is most likely dynamic, as a static IP address (one that doesn't change) costs extra money and is usually for people who host websites or other services from home. However, I have a dynamic address (technically speaking) from Comcast, but my IP address changes once or twice a year at most since the modem is always connected.
Getting a new IP address is as simple as unplugging your modem (cable or DSL) for a few minutes, although it might not always work this way. You may have to leave it unplugged longer. For example, one time the power went out for a couple minutes and I had a new IP address afterwards. Strangely though, one time I went on vacation for a week with the modem unplugged and when I plugged it back in I still had the same IP address.
If you still have dialup, you get a new IP address every time you dial in.
Hope this helps!