CMYK to RGB - RBG to CMYK conversions

Depends I guess on whether you're planning to print a couple of shirts to test, or a thousand to sell...  if it really is print-on-demand (customer wants one shirt, they print one shirt) then the waste should be small. Fabrics and printing techniques vary enormously, so I'd go for the swatches if it were me. Perhaps with a fave design on one tee, at the same time.

I wasn't trying to be sarcastic, we hear constantly from people who assume that buying pro software will give them the ability to pro work at pro quality. We constantly hear that "Photoshop is broken" because it doesn't do what was wanted after five minutes random clicking. It amounts to buying new carpentry tools, and blaming them when your first furniture is a bit shaky. I wouldn't try to put anyone off, so long as they have the time and dedication to give to the task. As a new business founder you're probably full of energy, which is a great start. You might consider graphic design training, there's a lot online now but be sure to find reputable info (free is often worth its price). If on the other hand you just want (for example) witty political messages and a black and white cartoon, you may not find it's worth getting deeply into this. You know your customers expectations better than we do!

Calibrating your monitor is about MAKING a profile for the monitor. Once you have that in the system, colour managed apps (which is less than you'd think) are able to show accurate colour. I'd like to hear recommendations rather than make them.

By the way, did the company give any kind of swatch (printed or fabric)? It would be nice to know if they have any reference point and what they say on them.

Yes, I'm saying Illustrator will convert to an RGB PNG because there is no such thing as a CMYK PNG. It will do this using options you didn't choose. If you have trouble believing it would do such a thing (it is surprising), open the PNG in Photoshop, and use Image > Mode to check. When you export, it doesn't change the original at all (export is not like Save as) so nothing changes on screen.

You may not be a pro designer but your questions indicate you want to get the RESULTS a pro designer does, so you need to learn that trade. It isn't about clicking the right box and suddenly you have all the skills you need... Sorry, but that is part of the business you really need to know: just like if you open a repair garage you need to know how to repair cars, or employ someone who does. Calibrating your monitor is a vital first step, without it nothing will match except by accident. (Accidents are the worst, because you can get a long way with happy accidents, and then have a really big fall).

It may also be useful to consider that you're printing garments, not coffee table books. How good do this company offer to deliver? Are most of their customers happy if their reddish colours are some kind of red, or are they offering fine art reproductions? Costs will vary radically, so will quality. The very fact that they only take PNG suggests they are consumer level commodity printer with no great expectations of quality (I do not know their reputation).

You ask "just use the internet to look for the RGB color values". Absolutely not. That follows from the idea that "this RGB colour #123456 is a standard colour you can use and it will be the same everywhere". This is completely wrong, but the internet is full of people repeating this nonsense. Unfortunately nonsense on the internet is one of the biggest obstacles to learning how to do colour correctly. Knowing how to make a Youtube video doesn't mean someone knows anything about anything!

CMYK is one way of mixing inks for colour printing. There are many others. For example 6-colour printing is often used. Very simple printing may just use CMY. For garments it's often necessary to start with printing white ink. Some providers may select spot colours as needed. All of these special things aren't done with CMYK, and so you wouldn't be working in the same colour model as them at all, if that's the case. Some garments may absorb huge amounts of ink too, so you might need to allow a dot gain that is out of the ball park; depends on the technology. This is why you should let them do the conversion; and why they will just convert your CMYK back to RGB before using the own setups. By choosing CMYK you are making colour matching virtually impossible.

You in any case have to send PNG, I think? There is no such thing as a CMYK PNG, so when you saved PNG it would be converted - perhaps without your knowledge, and without the chance to choose the correct settings. This would be risky indeed.

Colour matching RGB is a matter of using colour management correctly. This includes working with a good quality monitor which you have calibrated, the correct selection of profiles, and correct profile embedding choices; knowledge of gamut issues and how to prevent them. You have the secondary issue of previewing things on a garment that isn't white; this needs special expertise and cooperation with the provider, because it won't be the same as previewing on coloured paper.

No, it won't convert to the CMYK values you worked with, that's not how it works. You should note that RGB is now the preferred way of colour design for many  professionals, who do not use swatches. Instead, they rely on colour management, the careful use of good quality hardware, software and settings so that what you see on screen matches what you print. Designing in CMYK may be completely wrong for this workflow: they might not even use CMYK inks to print on a garment.

This doesn't sound too comfortable.

Since nobody knows how the printer will be doing the conversion, you can't be sure. You'd better do some test rounds with this printer.