I have been a long-term member on two internet forum websites.  I have enough reasonable evidence to suspect that someone from one website forum followed me to the other.  I won't go into detail why or how it was done, but the point remains is that there is a high probability of my suspicions being correct.  Now, I would like to know how I can find out someone's IP address, so that I can see if the two IP addresses, one from each forum, match.  In what way would it be possible for me to find this out?  Thank you.

Even then, depending on your location, or that of the forums in question it may be illegal for the sys-admins to provide you with the personal details of another member. Law enforcement would be your only avenue, and then only if you can provide evidence that a crime has been, or is being, committed.

Even if the two IP addresses match, that still wouldn't prove that the actual user is the same person. Proxies, corporate gateways, etc. could all play a part.

Not sure why someone following you from one forum to another would bother you. If that person is harassing you with posts, explain the situation to the site admin and he should be happy to ban them.

send him a PM with an image hosted on your own server,
check the logs