Up to this point, all I've ever done is to design something on the screen....But, now I need to get things printed from a printer shop. Some advice I've had is that I will need to supply Pantone Colours and/or CMYK values.
When designing for print, don't think so much in terms of "color." Think in terms of inks. Printing is a very physical/mechanical real-world endeavor, and much of it boils down to practical common sense.

One of the first things you need to consider when designing for print is how many inks will be used to print the job. You should make that decision up-front.

First, Pantone is just a company. It's a company which began in the printing industry long ago (long before graphics computers), and which developed a scheme for mixing a set of its own specifically-colored inks (spot colors) from a smaller set of actual ink pigments. It publishes its formulae for mixing the smaller set of "base" inks in order to achieve the larger set of mixed inks with reasonable consistency. Then, for use as selection references, it published printed swatch books which were actually printed using those inks. It's all proprietary. Yes, it's widely used, but it's not some kind of universal scientific colorimetrc rule of nature.

The original concept was simple: Assuming print shops A and B both buy their inks, X and Y, from Pantone, and assuming they use the same published proportions to mix up a third ink, Z, then the results from print shop A and print shop B should be the same color, even though they are different print jobs, run on different presses for different customers in different locations.

But that all assumes spot color; a term which means you're going to load the actual desired color of ink (Z) into the press.

So basically, you only need to worry about "Pantone" if you are actually going to print using spot color inks; in other words, you are going to actually load the press with the color(s) of ink you want to be the end result. Think of the ubiqutous US politician's calling cards and signs. They're almost always red, white, and blue. The paper is white, so you can print the job by loading a red spot color ink and a blue spot color ink into a press which has only two print heads. Use the same brand and color of inks, and you can print half of the cards in California and the other half in Georgia and they will be (within reason) the same color.

But that's only practical for up to a few inks. A printing press only has a few inkwells into which to load ink. You can't load an infinite number of different-colored inks into a press in order to print, say, an ordinary full-color photograph of the same politician's toothy grin. That's where process color comes in.

You resort to printing with a set of four "primary colors" of inks in such a way that when overprinting each other, they visually blend together to simulate a broad range of colors which can reasonably represent all the colors in the photo. If you're project is going to be printed using this "four color process," you don't really need to concern yourself with "Pantone" at all.

So later, predictably, the desire arose for being able to reasonably simulate a particular Pantone spot color ink when printing using four color process. So Pantone began publishing its "official" recommendations for CMYK mixes which approximate the colors of its spot color inks. Again, all this predates computers. When graphics computers became commonplace, Pantone began publishing software versions of its ink color references. And since then, many digital designers insist on using Pantone swatches as their primary reference, even when designing for media (signage, for one example) which don't even have colors corresponding to Pantone's.

If designing for four color process, there's nothing that says you have to refer to Pantone's proprietary colors at all. If you want "red" well, you just can't get any more "red" in CMYK than 100% Magenta, 100% Yellow. But I guarantee you that will fall far short of the intense "redness" of the Mustang I had occasion to rent on a particular trip to San Diego.

Does that mean it's impossible to print a suitable photo of that Mustang? Nope. That's where color correction comes into play as opposed to mere color calibration. Color correction throws the spectrophotometer out the window, takes into account the undeniable and unavoidable limitations of CMYK four color process, and intelligently exploits the perceptual and contextual adaptiveness of human vision to make a CMYK print more convincingly suggest the brilliance of that Mustang. And that's where continous study, observation, and experience comes into play.

OK. It seems that I need to buy some swatches.
Maybe. Maybe not. (At least, maybe not right this minute.) Again, it depends on what exactly you're needing to print, and despite the length of this thread you really haven't yet shared one bit of information about that.

Not all printing is as all-fired color-critical as a national ad campaign or sales brochure for a very red Ford Mustang. In fact, truth be told, most isn't. And color-calibration, while not the highest ideal panacea which many of its devotees mistake it to be, is also not completely valueless. Device manufacturers and software publishers have gone to great calibration lengths for decades to at least keep you in the colorimetric ballpark by default.

Most projects for which beginners in print are likely to be hired are probably not in the realm of table-top glitzo color-critical. If the project you're working on is, for example, being commissioned by National Geographic, you should say so now. Otherwise, while getting your feet wet in ink-on-paper, you will probably come out just fine even committing the "unspeakable travesty" of sending an RGB PDF for automatic in-RIP converstion to CMYK for offset print.

But if you do that, just don't come on to the printing house like a super-sensitive artistic savant complaining about the lack of "brilliance" or "punch" or "good humor" or any other atsy-tartsy ill-defined term. Trust the printing professionals to take care of you. Seek opportunities to visit (or even work in) the press room. Read, study, gain experience.

And yes, when you can, buy a Pantone swatch book. You'll probably need several. And while you're at it, a PostScript process swatch book.

And don't be afraid to do some methodical experimentation of your own. For example, when you print the next process color project, place a few squares of incremented CMYK values in the waste area of the press sheet to test what you consider "the best" match for a particular spot color ink that will be used in the next upcoming project.

That kind of thing can accellerate your learning. It's not difficult (especially in digital print-on-demand environments) to justify printing a sheet or two of your own color swatches just to zero in on the best process specs for a particular customer's spot-colors.

Then is it a case of trying to see which colour closely matches what I'm looking at on my screen?
Never trust what you see on your screen. Yes, go to reasonable lengths to have a reasonably well calibrated screen. But always take what your screen shows with a health grain of salt. Don't believe me? For just one kind of consideration, do the following:

Turn off your monitor. Do it now.

What color is it? About 80% gray?

Now I ask you: Even with all the color-calibration you can muster, what's the darkest black your monitor can display?

See what I mean?

So sure, you want your monitor to look reasonably like print; not bluish, not yellowish. You want your several graphics apps to display colors with reasonable similarity. But when it all comes down to it, if you're ever going to be serious about this stuff you have to learn to fly by the numbers, not just the appearance of your monitor.

Example: An enterprising young illustrator sits in his mood-enhancing darkend room and does absolutely killer graphics which he hopes to someday see published in the game industry. He sends one of his stunning works to have a one-off poster printed for his portfolio. And when it comes back, it is utterly flat and featureless in the all-important shadow areas. That's just one kind of the ugly surprises in store for you if you don't seek to understand print-centric CMYK values.

I just thought that there might a conversion, of one type to another?!
As you've seen already in this thread, "conversions" abound....

To be able to give exact colour names/numbers.
...but the operative word here--and the falacy--is "exact." Not when substituting RGB for CMYK. And even when you achieve it (as many claim by using Lab as the intermediary color space) hair-splitting colorimetric calibration is not really the most important consideration for good color.

And not what the eye may be able to match or not.
You've got to develop a "mind's eye" for actual ink-on-paper, not just for glowing backlit LCDs.

JET