Tl;dr

If the instructions below are too much to read, you must not file a bug in the issue tracker but instead use your distribution's bug tracker or support forum.

Preliminaries

First take a look at the list of Frequently Asked Questions to see if it contains the answer to your problem.
The issue tracker is not a support forum; for support questions use only your distribution's support forums!
Adherence to the GNOME Code of Conduct is mandatory in the GNOME issue tracker.
Read these instructions on how to write a good bug report.

Feature requests

If you want to request a feature, please first look through the list of already-reported tasks in the issue tracker, and also the list of closed tasks in the issue tracker.

If a request for the feature you want hasn't been filed yet, you can file one against GNOME Terminal for the user interface, or against VTE for the terminal emulation.

If a request has already been filed, you can CC yourself to track its progress. If a request has been filed but rejected, and you have good arguments refuting the arguments made against it in the report, you may file a new request and link to the old request, detailing your reasons. Do not add comments to closed bugs however, since they will most likely just be lost.

Bugs

1. Where you should report the bug to

If you're a developer of GNOME Terminal, VTE, or GNOME in general, you can file the bug in the GNOME issue tracker. In all other cases, you should file the bug against your Linux distribution's bug tracker; see here for a list of distribution bug trackers. The distribution bug triagers will then gather all the necessary information and confirmation before forwarding the bug to our issue tracker, if necessary.

2. How to proceed

First you need to make sure the bug really exists in GNOME Terminal or VTE, and isn't caused by other things. Remember to always run a new instance of the GNOME Terminal server process! See here for detailed instructions how to do so.

If the bug is a graphics problem, or 'slowness' (esp. when using transparency), or a crash where the stack trace indicates the graphics driver, the X server, or graphics libraries (cairo, png, ...), then you must test if the bug happens when using a free software graphics driver, on an untainted kernel, using the GNOME default theme «Adwaita» and the default icon theme.
If the bug is a graphics problem, or a focus problem, or a window sizing problem, you must test with the default GNOME window manager «GNOME Shell» in click-to-focus mouse mode (e.g. focus-follows-mouse is not supported!).
Additionally, if you're running on ubuntu or its derivatives, you must test without the ubuntu modules; see here for instructions.
If the bug is an internationalisation or localisation problem, make sure you use an UTF-8 locale; legacy locales are not supported.

If all these tests have been done, and you can reproduce the problem, you should first search if there is an existing bug report for this problem under both gnome-terminal and vte. If there is, append your information to the existing bug; otherwise file a new bug against GNOME Terminal for the user interface or against VTE for the terminal emulation.

3. What information to provide

Make sure to clearly document all the steps necessary to reproduce the bug.
Specify which distribution your're using, and which version.
Specify the exact versions of gnome-terminal, and vte.
If the bug is a crash, include a full stack trace for all threads, with debug info installed. See here for detailed instructions. Also search your distribution's support site for specific instructions, in case they vary from the general guidelines.

4. Security sensitive bugs

If you think that the bug might have security implications, you should follow https://security.gnome.org/ instead.

5. Forwarding bugs from other bug trackers

If you're forwarding a bug from another bug tracker (e.g. a distribution's bug tracker), you should link to it, but also include all relevant information from the other bug; do not just link to the bug and expect the GNOME Terminal developers to create an account there to look at it, or to go looking through tons of potentially irrelevant information in the downstream bug tracker!