Pebs, this is a common problem. Check out programs like Supernanny or Little Angels or other tv programs that deal with parenting tips.
At 14mo your ds should be able to be calm and reasonable most of the time (provided he is not ill/teething etc). If the problem relates to while you are in the kitchen, can you get him something to do in the kitchen? Does he like to play with noisey pots and pans? Does he like to pour water from a jug into a cup (and back again)? Does he like to sit on the sink and fiddle with the taps? (with you nearby to ensure no falling off). I found all of the above really captured ds's attention when I was in the kitchen, and this helped to let him get on with his playing while I cook. I also let him "help" me (stir things etc - even if I had to put some flour in a bowl of water for him to stir and tell him he was "helping") and then give him a cup of water which he stirs endlessly. Let him help unpack the shopping. Sometimes my ds likes to unpack some of my groceries from my cupboard. It keeps him busy, and I can get on with my cooking, then he "helps" me repack the cupboard at the end.
Problem is that if you are picking them up all the time, then they will continue the behaviour because they get a "reward" for it (that is, your attention).
So I suggest, get him involved if possible, or distract/ignore if you can. Otherwise, you need to get tough by introducing time out (even if only for 30 seconds or so). Explain clearly why you are putting him in timeout (I need you to stop whining), and what to do to avoid timeout (there is no time out when you are happy and smiling). Of course, the alternative is just to put up with the behaviour.
My ds never went through a phase like this, although I am sure he would love to whine for days on end if we let him. He is now 2.5yo. Every now and then he gets a bit whiney, and we tell him that is not allowed in our house. We have used timeout since he was a year old. He has not had to have much timeout, but he knows that whining does not get him anything good (no attention or anything else he wants). He also knows that if he puts on his "happy smiley face" (even when he does nto feel like it) he is much more likely to get what he wants. He is a very happy cheerful chap and we include him as much as possible in daily activities, and have done so from early on. TBH, I did not have the energy to deal with a screaming child, and although he does push his luck, a short timeout usually sorts him out for months to come. He is a pleasure at home and out shopping etc. It is difficult to be consistent, but it really does pay off. Seriously, watch those supernanny tv programs - they have heaps of tips which really do work.