To remove the Estimate shipping and taxes block you'll need to customize the layout of your theme.
In order to do this you can copy app/design/frontend/base/default/layout/checkout.xml into app/design/frontend/default/grayscale/layout/ (you need to replace default/grayscale with the path to your theme files) and after that customize app/design/frontend/default/grayscale/layout/checkout.xml removing the following line there:
<block type="checkout/cart_shipping"
name="checkout.cart.shipping" as="shipping"
template="checkout/cart/shipping.phtml"/>
Another, more correct way (as it will not override whole default
layout, so will be more upgrade-friendly) would be to edit local.xml
file in app/design/frontend/default/grayscale/layout/ and
add the following content right above the closing </layout> at
the bottom:
<checkout_cart_index>
<reference name="content">
<block name="checkout.cart">
<remove name="checkout.cart.shipping"/>
</block>
</reference>
</checkout_cart_index>
After that you need to refresh Layouts cache at System > Cache Management
