There's a reason beginning film photography classes start with:

• 35mm B&W film.
• Normal exposure and development.

When you know what's normal, you can sort out what factors cause what effects when you experiment with different film types, development processes and formats.

what matters most when trying to get this retro look?

Anything shot on "retro" film with "retro" glass will have a "retro" look because that's how they were originally shot. That still covers a huge range of different looks, so you'll have to experiment to hone in on what you want.

Does the camera matter at all?

In film photography, the camera is just a light-tight box to house the film and attach a lens, so it usually doesn't matter much, as long as you can use the lens you want. When a less-than-light-tight box is desired (lomography), camera choice matters a bit more.

Should I look for specific lenses or does that even matter?

Different lenses do have different characteristics. Some lens designs produce distinctive images (triangular bokeh, strong swirl, etc). If this is what you seek, lens choice may matter quite a bit.

Is it all about the film (certain types, temperature, ISO) in this situation, should I learn about specific techniques in the darkroom (push/pull processing?) that creates this grain, or is it about the way that I shoot the photo (shutter speed, aperture) that matters?

Some image characteristics are associated with the film used. Not just ISO, but the specific manufacturer and product. Film processing also matters (developer, push/pull). Camera settings (exposure) usually depends on the film selected and the type of processing desired (eg, under expose for push processing).

There are some rules of thumb:

• High ISO film ⇒ more grain and less contrast
• Push processing ⇒ more grain and contrast

However, there are film and development chemistries that are designed to behave differently. Kahovius explains a bit more.

Paper types and processing also influence results. If you intend to print your work, you may be able to achieve the looks you want without changing exposure and film processing. This is equivalent to post processing digital photos.

My recommendation is to start with learning normal first. Then experiment to figure out your preferences and the effects different factors have.