No doubt that is sound advice from the legend that is Kayakamina.
I will also have a go at some of those others that he mentioned now.
I write as someone who was, dare I say, until recetly very similar to you;
I only started the Sunday Times a couple of years ago, and that was all I had time to do. I then did the Sunday Times crossword club monthly special ( harder and more obscure words ). Then after e-mailing a setter about a clue in the sunday times that seemed to break the 'cyrptic rules' learnt that the Sunday Times is somewhat very 'laissez faire' about its clues.
So I started the Saturday times a few months ago and the cyptic Jumbo which are better in this respect and then just plunged into having a go at the Listener - I first looked at the answers to the previous weeks to get a feel and then head down. I don't have time or the experience to always complete them and I am a million miles away from the speed and knowledge of guys like K and the other brainiacs on this site but I have now completed three and now also do the spectator and mephisto which I'm sure I would still be struggling with had I not struggled with the Listener first.
The big benefit to having a go at the Listener is that it tends to be very tough but very precise and fair, as well as great fun. BAT by Scahenfreude last week being a great example.
The point I am clumsily making is that, hard as it is, diving into the deep end forces you to raise your game as it were. I would enourage you to have a go.