You could do something like this.
Add a custom catalog/homecategories.phtml
file and add it in your local.xml
.
homecategories.phtml
$helper = Mage::helper('catalog/category');
/** @var $helper Mage_Catalog_Helper_Category */
$topCategories = $helper->getStoreCategories(true);
?>
<?php if (!empty($topCategories)): ?>
<div class="home-categories">
<ul class="level-1">
<?php foreach ($topCategories as $topCategory): ?>
<?php /** @var $topCategory \Mage_Catalog_Model_Category */ ?>
<li>
<a href="<?php echo $helper->getCategoryUrl($topCategory); ?>"><?php echo $topCategory->getName(); ?></a>
<?php if ($topCategory->hasChildren()): ?>
<ul class="level-2">
<?php foreach ($topCategory->getCategories($topCategory, 0, true) as $category): ?>
<?php /** @var $category \Mage_Catalog_Model_Category */ ?>
<li>
<a href="<?php echo $helper->getCategoryUrl($category); ?>"><?php echo $category->getName(); ?></a>
<a href="<?php echo $helper->getCategoryUrl($category); ?>">
<img src="<?php echo $category->getImageUrl();?>" alt="" />
</a>
</li>
<?php endforeach; ?>
</ul>
<?php endif; ?>
</li>
<?php endforeach; ?>
</ul>
</div>
<?php endif; ?>
<?php
local.xml
<cms_index_index>
<reference name="content">
<block type="core/template" name="home-categories" as="home_categories" before="-">
<action method="setTemplate"><template>catalog/homecategories.phtml</template></action>
</block>
</reference>
</cms_index_index>