Microsoft publisher is a reasonable package for designing flyers in.
You just need to make sure the bleed is factored into the page size when creating the page, and that the export properties are correct.
For example, an A5 (148mm x 210mm) flyer should be created as a 154x216mm document, to allow 3mm bleed on all sides. your safe area should then be 8mm in from each edge.
For help with bleed, please see our explanation here.
We have templates of common formats (including Publisher) for download here.
Once you have created your flyer in publisher and have made sure it has the correct bleed on it, you should export it as a PDF as the best file for us to print from.
So click, File, Save As.., .PDF and select commercial press settings as below -