I think you have a pretty good plan.

Camera: Your smart phone camera should be a decent way to start. I think it's important that you learn to use the settings of your S10 and shoot in raw. It'll be important to get that "non typical smartphone look" you mentioned. You'll need a workflow for post production as well.

Tripod: I'm thinking something like this would would be a good start. It would allow you to shoot from above. You can also try just holding the phone from above. That would give you the most control.

Lighting: Those lights are probably an ok start. Since jewelry is usually reflective, you typically don't want to shine a light directly at it. I would pick up something like this. You'd position the diffusion panel between the jewelry and the light source and it will have a nice diffused effect. You want the jewelry to reflect the diffusion panel, not the bare light source. I think you could also get by with using natural window light, especially if you can diffuse the light with a translucent curtain.

Surfaces and Background: From your example images, it looks like most could be shot from above. I would grab a set of those backgrounds from Amazon and start sourcing more materials. The ones with the printed textures are very popular. I would also recommend tile samples, fabrics, wood panels, and fancy art store paper. Just start collecting interesting surfaces.

Additional Lens: I've always thought those smart phone lenses were a gimmick, so I have no input there. I wouldn't be in a rush to add it though.