I'm looking for a way to turn an ordinary flat font into a 3D metallic text, kind of like the Game of Thrones logo here, or the Skyrim text here? Or this logo here. And maybe even any hand drawn shape, not just text? It doesn't need to have these additional effects like scratches and imperfections, just the 3D metallic stuff. It would be especially helpful if it was possible to do in some kind of free software (GIMP, Krita, Inkscape, etc).
Metallic writing is easiest in Photoshop because it has layer style Bevel&Emboss which your examples seem to use. It alone gives numerous variations. You'll find easily numerous gold making receipes which use that effect. In addition they insert color variations for greater plausibility - gold gradients or even a photo of gold to be used instead of a single solid color fill.
Glossy chrome doesn't need colors. You can well start from grey writing. But to make it glossy it needs heavily distorted environment reflections. Highly polished gold needs as well color and reflections. I skip creating realistic reflections. In small items one doesn't expect visible reflected environment details, gradients are often enough. Bevel&Emboss itself can create enough lightness variations for many purposes - just one example: Here's white text. With strong chiseled Bevel&Emboss it has quite extreme contrast which jump out on a greyish background:
This sample was from Photoshop. It has also a slight drop shadow.
You wanted free tools. Photoshop costs money, but that's no problem. There's (still) free web service https://www.photopea.com which imitates slowly but otherwise succesfully many Photoshop's functions. Layer style Bevel&Emboss is one of them. We can try to make gold with it:
Bevel&Emboss for fake 3D appearance works well, but this single yellow text color doesn't make it. It looks too green. Turn it to more red, then it looks too orange. For better results cut the text from a golden gradient or from a photo of gold. Even some blurred random gold samples can be good. Here's an attempt to cut the text from a blurred colored piece and apply Bevel&Emboss:
One can make a selection with text by clicking the text icon in the layers panel and holding Ctrl at the same time. Here the selection was used to cut a piece of color. That color was pasted as a new layer and it got Bevel&Emboss. Whitening and darkening opacities are turned to much less than 100% to keep the colors visible.
Of course this works with any shapes, it's not "text only". It makes the borders between color and transparency look extruded to 3D and rounded or chiseled. Photopea hasn't custom edge contour curves like Photoshop, but there are still many usable presets.
The Bevel&Emboss dialog is complex. To use it maximally one must understand every setting. Adjusting the light is especially tricky and depends heavily on the geometry. Very easily something is too dark or too bright or there's no difference where one expects a high contrast.
GIMP's basic package doesn't have Bevel&Emboss, but it can be found as a Script-Fu extension. Here's an example how to use it: https://www.youtube.com/watch?v=EVVlItASBoM .
BTW. The gold is blurred colored random dots. Their colors are from different pieces of gold in a gold mining television program.