Publising Information on Your Key
Hiding your private key is obvious, and you already noted that.
The only further thing not to publish (without intent to do so) is the revocation certificate you might have pregenerated (and GnuPG 2.1 is doing so by default for a good reason). With such a revocation certificate, others could revoke your key by uploading it to the key servers. Sharing with a trusted friend or family member on the other hand might be a good thing to do: they could still revoke your key in case anything happens to you (consider it a kind of digital last will).
Other things are hardly a problem. If you publish your public key (or fingerprint, mail address, user ID: all of those could be used to retrieve your key from a key server, given it is uploaded; and if you use your key, expect it to get uploaded by anybody by mistake or intent). This might allow to connect your Stack Exchange account with your real identity (which on the other hand does not seem to be a problem to you, looking at your profile).
From all further information like key IDs, fingerprints, user IDs, ... neither your private key nor revocation certificate can be calculated. Key IDs are the lower parts of the fingerprint, which again is calculated from the creation time of your key and public key.
Publishing Information Created Using Your Key
If you do not want to reveal your identity, be aware when publishing encrypted and/or signed data for/from your key, even if your strip parts of the ASCII-armored or binary output; or publish parts from your public key. It includes the whole key fingerprint, and does so in the very first parts. If you have something like
-----BEGIN PGP MESSAGE-----
Version: GnuPG v1
[snipped some lines for assumed privacy]
-----END PGP MESSAGE-----
an attacker will still be able to parse the packets manually (although neither
gpg --list-packets nor
pgpdump will help at analyzing such broken information without modifying their code), and you need some understanding of RFC 4880 (and a hex editor).