Which step/component of film photography matters most when trying to get a very vintage, grainy look?

Get high ASA (that gives you larger film grain) and underexpose (that makes the grain most visible). It's actually not all that different from digital in that respect, except that the digital results are not as much grainier than they are noisier.

To expand on what user95069 said, it's primarily about the film being used, not the camera. I've taken multiple photos on different cameras on various film stocks, from budget point-and-shoots to automatic SLRs, and the primary factor when it comes to grain is the film and how it's exposed, and the type of camera and lens used has no visible effect on grain.

To increase grain on your photos, here are the four most important techniques:

• Use a higher-speed film. A higher film speed ISO/ASA rating will increase the visible amount of grain, as higher-speed films need to have larger grains so that they're more sensitive to light, and this leads to higher levels of grain on the images you get. Take a look at the datasheets for Kodak's current line of professional color negative films, and you'll see that the print grain index, a scale of visible grain, increases with film speed, with Ektar 100 having an index of <25 (imperceptible grain), while Portra 400 and 800 have 37 and 48 respectively (with a 35mm camera on a 4x6 inch print).

• Use a smaller negative (35mm or lower). I noticed your comment on their answer that you're planning to get a medium format camera. I'd suggest you use a 35mm camera instead if your main focus is on grain. The amount of grains per unit of length is the same regardless of negative size, so having a smaller negative will increase the proportion of visible grain, and this is reflected in the datasheets I linked above. As an example, Portra 400 has a grain index of 37 with 35mm, but 25 with 6x6 cm medium format (on a 4x6 inch print).

• Use a consumer film rather than a professional one. Budget films tend to have grainier formulations, as they use cheaper emulsions that aren't as sensitive to light, and so require more grain to achieve the same speed. As an example, the professional Portra 400 has a print grain index of 37 (as I pointed out before), but the consumer-level Ultramax 400 has an index of 46 (again, with 35mm film on a 4x6 print).

• Underexpose and push your film. This makes the grain more visible, as lower-speed films contain fewer grains which are optimized for a lower speed, which are now doing more "work" so to speak. As an example, I shot this roll using Ilford HP5 Plus 400 pushed to 800 despite it being a clear day, and the grain is more accentuated.

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.

If by vintage look you mean grain specifically, then the following are all that matters:

• Emulsion. Faster films (higher ASA/ISO/etc.) give more grain than slower films due to the way they're manufactured.
• Developing solution. The chemistry you develop the film in has an effect on the resulting grain. There are acutance-enhancing developers, such as Rodinal, which increase edge sharpness. This has the effect of making the grain structure more pronounced. A "fine grain" developer such as Ilford Perceptol works on the opposite premise by dissolving the edges.
• Amount of development. The longer you develop (or the higher the temperature of your developer, or the less diluted your developer), the more silver halides in the emulsion are reduced to metallic silver. Grain becomes more apparent. This is why "pushing" – i.e. underexposing and overdeveloping – leads to a more grainy look. (It has other effects as well, such as increased contrast.)
• Size of negative. A fleck of grain is much more prominent if the negative is small (e.g. 35mm camera or smaller), since it is larger in proportion to the overall image. Medium and large format "suffer" from less grain, relatively speaking, for this reason.

For grain, it doesn't matter what camera or lens you use (assuming constant negative area).

or link me to a good resource on this!

Rather than looking on the internet, I would buy a couple of second-hand books, which contain a great deal more information than you can easily find in any one (or even two) place on the internet (for example, Henry has 40 pages on grain alone). In a rough order of increasing sophistication/nerdiness:

1. Michael Langford, The Darkroom Handbook
2. Ansel Adams, The Negative
3. Barry Thornton, Edge of Darkness
4. Michael Langford, Basic Photography
5. Richard Henry, "Controls in Black and White Photography"