I run a forum website in English and want to expand into other languages. (Spanish, German & Chinese)

The forum software has language packs that translate the nuts and bolts of the forum administration/user preferences, etc... but the actual content is user-generated.

Should I simply create new forums in the new language?
Do I need to buy other top level domains like .es, .de and .cn ?
Do I set up new forums in each of the new TLDs, or redirect the ccTLD to the main .com, and have the foreign-language forums listed on the main .com forum index page?

no need to buy new domains. Google recommend either:

es.forumname.com
or
forumname.com/es/

it really just depends on which is easiest for you.

Best treat the index page as a global entry to your site, then split them off into the other languages from there.

I've never seen this done for a forum before, I'd love to see the result so please publish a link once it's done

This sure would be a tough call. I think you should consult the forum software vendor before make any decision on this.

Alright, thanks for your help. It seems the most important part is the content, and I can worry about subdomains vs. TLD later!