Designs are normally supplied in two different ways to printers these days - digital or hard copy. We'll concentrate on digital here but generally printers will accept hard copies (e.g. a design done on paper) that they can scan and then work from. Check first that your chosen printer will accept this and what the costs might be as it's likely more work will be involved in getting your design onto the tshirt.
Before starting your original tshirt design make sure the resolution is at least 200 pixels/inch or your finished tee may not come out as you imagined. Remember the default resolution of most graphic software, normally 72 pixels/inch, is intended for web graphics and not reproduction.
Provide as much information as you can with your design such as pantone colour numbers (if used) and the size and position of your graphics once on the tshirt. In some cases you might be able to download a blank tshirt template on which you can indicate the best position for your design.
Printers may have different requirements when it comes to fonts and what format to save your tshirt art in (e.g. eps, jpg, tif) so it's best to check first before sending in your designs. Once you have this information they should be able to accept designs sent on floppy disk, cd, or other digital storage methods (check first). You might be able to email your design or in some cases submit via an online contact form on their website.
Once a printer has your design they should be able to give you information on the most suitable printing method and a quote based on your requirements.