I know GPLv3 requires to "disclose the source".
But does that mean that I have to bundle the source code with my release?
The sources don't necessarily need to be bundled together with the binaries. The options you have are listed in Section 6 (Conveying Non-Source Forms). See Section 6 of the GPLv3.
Does the code need to be open on the Internet (Github), or a link to a ZIP archive as download, ...
The basic rule is that if you offer the binaries (or "object code" in the language of the GPLv3) on the Internet (either "gratis or for a charge"), then you must ensure that the Corresponding Source is also available online "at no further charge". See Subsection 6d for the details.
This means that the sources don't necessarily need to be "open" to the Internet (e.g. they could be behind a paywall), but it means that once you give someone access to your server to download the binaries, then you must also offer the same person access to the Corresponding Source at no additional charge.
The server that hosts the sources may be operated by a third party (e.g. GitHub) but you are responsible for ensuring that the sources remain available from that server. Also, in this case, you must provide "clear directions next to the object code saying where to find the Corresponding Source."
Section 6 also gives you non-Internet based ways to convey the sources, but only in some scenarios. For example, if you are distributing the binaries embedded in a physical product, then it's allowed that you distribute the Corresponding Source on a physical medium only (e.g. a CD, a DVD, a USB-stick, etc.) and that means that you wouldn't necessarily have to make it available online. See Subsections 6a and 6b for the details.
... or are you even allowed to specify "Email me to obtain the source code"?
Under the GPLv2, this option was generally allowed and it's called a written offer. Under the GPLv3, however, providing (only) a written offer is only allowed if you are distributing the software as part of a physical product, or if you received the software itself with a written offer only. See Subsections 6b and 6c for the details.