Cleaning the teats/udder pre-milking with either teat wipes (quite expensive if you only have one or two) or using unscented baby wipes (cheaper, but still not that environmentally friendly) or using warm water and cloth to wipe, then dry with dry cloth (cheap, probably more environmentally friendly, but not hygienic if milking more than a couple of goats)... is all that is needed. I teat-dip with a weak iodine solution after milking, as I had some mastitis issues a couple of years back and have since done it after every milking - a lot less mastitis, but still occasional problems. I just put some Capriclense and add a lot of water to the teat-dip cup, you want it quite weak for teat dipping.
You filter the milk through either proprietary milk filters (easy and quick) or some people use muslin. If the milk doesn't go through the filter easily - you may have a problem with mastitis (often sub-clinical, but it is a quick indicator that something is amiss).
I do not pasteurise the milk we drink fresh, although have had to pasteurise the milk I used for cheese over summer, due to (most likely) yeast and/or bacterial problems in the warm and humid weather.