When printing on the t-shirts of different sizes all you need are two size prints.
One for children-teen size and one for adults
For children-teens (size 6, 8, 10, 12, 16, S) it's enough that you have a print of 25x25 cm or 25x30 cm.
For adults (M, L, XL, XXL, XXXL) you will need a print in size of 28x28 cm or if you wish to have really large print that covers chest and stomach area, you can go with 32x32 cm or 32x42 cm.
Genuinely, you can even go larger, since standard screen printing frame (working area) is around 40 x 70 cm (also depends what on kind of squeegees they use).
Image would be prefer to be in PDF format, previously made in PS (or some other raster-based program) in 150 or 300 dpi resolution. That image can be in full color, and they will use process colors (CMYK) to print it onto t-shirt. Those are specially pre-made colors for printing through very fine mesh so the image will look as a photography. You just need to pay attention that CMYK colors are mostly dull and darker than RGB, so always do the CMYK preview before exporting RGB image as CMYK PDF.
Standard screen-printing carousel has 6 working stations, so if you are making an image in some vector program (CorelDraw or Adobe Illustrator) you might want to omit your design to 6 colors. Do this only if you are preparing vector image, although, you can always export vector image as raster image, in which case you really don't need to omit colors or anything since they will be printed through fine mesh as a photography (again process colors used).
This really depends on what kind of design are you making: clean, simple design or full-color (lots of colors, shadings, tones) photography.
In any case, as long as the image is larger (or same size) than A4 paper format and it is in 150-300 dpi resolution, it will be fine. Anything below that can turn 'blurry' when screen printed as a full-color design, photo.
For vectors, doesn't really mater what size the image is since vectors are scale-able and they don't lose quality with re-sizing.
So, PDF, 150-300 dpi, working CMYK prepared in sizes I wrote above.
If you exported your vector image as PDF it will remain vectorized upon importing into other vector programs prior printing.
It would be good to add a preview of a t-shirt with your design so ppl that will print it will know what colors to use (they can mimic colors from the preview you gave them, they can mix colors they have to make ones you show on preview) and on which positions to put your prints (shoulder, left side of chest, plain chest, back, etc)
If they use digital print-on-textile machine then it's all the same, just keep your design in 150-300 dpi resolution and it will be fine. Digital print-on-textile machines have a standard span of 1 m of working surface, so basically you can make a print that is going over the whole t-shirt. They print the textile, cut it and sew the t-shirt, so the whole surface can be 'enveloped' in your print. But, idk, if they are small company, they probably do basic screen printing + transfer prints.
Transfer print is print made on special paper and it is covered with powder glue. It's also done through mesh, and it can have 6+ colors or it can be full-color image (4 process colors, CMYK). They fix the image from that paper under the special heat-press onto t-shirt. In that case, size of the design is omitted by the size of the heat-press (standard size is 35x35 cm).
I hope this wall of text helps.
I'm working in the t-shirt printing company for ages so this is the everyday stuff for me.
We do all three, screen printing, transfer printing and digital textile printing + embroidery.
Regularly, when i find some time I make t-shirts for myself (and some friends) so I already have Toribash t-shirt, World of Tanks t-shirt, Portal 2, Call of Duty, and many more... Don't need to go to store to buy it, when I want one I made myself one. hehe...
