1. open [catalog]includes/classes/shipping.php for edit. Look for at around line 112 & 115 these two lines of code:
$cheapest = $rates[$i];and change both to:
$cheapest = $rates;
2. in admin, modules - shipping, give/change the sort order of the shipping method you want to set as default to 1, ie, the top most, or you will need to the '0' in
$ratesas shown above to a number that corresponds to your chosen shipping method's sort order, eg, 1 for sort order 2.
No other change is needed.