When doing an overview of the kitchen, the first place you want to go to is the sink. Run the water and check the temperature of the hot and cold water. You can also plug the drain, filling it up and then unplug it to let it drain out to see how the drains are working in that area of the house, if there are any strange sounds or if the water wasn't draining quick enough, you want to take a mental note of it. Check to see if the water seems to be flowing freely.
Also, check under the sink, are there any leaking pipes? Is there any sign of water damage? Is there a dishwasher next to the sink? If so, do the walls of the cupboard look they have been damaged by water?
A dishwasher is a desirable feature to the kitchen but if they are old or if they haven't been installed properly it could mean that water is flooding the kitchen floor or there is a leak in the inner walls. This would lead to the growth of mold or rotting floorboards.
These are just some basics when walking through the kitchen of a home for the first time.