Did you create categories correct, It must sub of Root categories.
You can use build in widget “Catalog Category Link”.
- Open your magento backend
- Navigate to CMS->Page
- Select “Home Page”
- Choose tab “Content”
- Click on button “Insert Widget…”
- Choose widget “Catalog Category Link” from list of available widgets.
- Fill widget required information and click on “Insert Widget” button
- Repeat step 5-7 for each category that your need to display on home page