The cart display is looking good, nice work. Now to get your shipping charge rules in place you need to edit the current cart object. To bring up the cart object interface you can go to Windows > eCart Object Panel, this will bring up the eCart object panel, just select the cart and click edit.
You will add a new rule to the shipping tab. In here you just create a new shipping rule for your first weight range. Give the rule a name, like 10kg or less, then add a condition to the trigger. The first condition will be based on subtotal of a column, then you will specify the totalweight column like this:
If the subtotal of the column totalweight is > 0
Next you will add in a second condition for this trigger. The next condition will use AND as the separator and will look almost identical to this one but you will be setting the upper limit for your first flat rate charge. So if you have a rate you would like to apply to all orders that are 10kg or less the second condition would look like this:
AND If the subtotal of the column totalweight is <= 10
For the calculation you would select flat rate and enter the amount you would like to charge for this range. This rule would then read, if the subtotal of the totalweight column is greater than 0kg, and less then or equal to 10kg apply the flat rate charge for the shipping.
You would repeat this for each weight range you would like a flat rate charge to apply for.