To embed fonts on your artwork, create your print-ready PDF using the setting PDF/X-1a as your default. This should embed your fonts. To check your PDF has embedded fonts, open your PDF in Adobe Acrobat and click File - Properties - Fonts tab, and each font should say 'embedded subset'.
If one or more fonts still aren't embedded then there may be a licensing restriction which is preventing this. In this case, you have two options. Either open your design file and convert to curves / paths / outlines, or flatten the layers in Photoshop. Remember to save as a PDF set at CMYK and 300dpi.