new biker

The only way to get better at riding is riding more.  Seriously.  You can say that you ride 2-3 times a week now, same as you do running/swimming, but you already have a running/swimming base so can maintain that base with that time of training.  Seeing as you just picked up the bike, yes, you have to ride more, and yes it is that much different. I started cycling in fall '07 and remember my first 30+ mile ride and averaging something like 15mph.  I was sore, I couldn't stay aero much (I have a tri bike), and yet could easily go out and run 15+ miles.  I started cycling, and come the spring/summer of '08 I was comfortable with 60-65 miles and my speed had greatly increased.  Come Fall '08 I was able to do a half-ironman (56 miles) with just over a 20mph pace.  This year I've been training for Ironman Florida in November and have already put nearly the same amount of miles on my bike that I put on all of last year...three weeks ago I did a 100 mile ride (a very flat 100) and averaged 20mph for the whole ride, stayed aero for the vast majority of it, and felt awesome when I was done.  BUT, if you look at my logs and the amount of miles I put in last year (about 3800) and the amount I've put in this years (nearly 3800 and still going) you'll see that it took A LOT of time in the saddle for me to get faster and get faster at longer distances.  Truly, the way to become a better cyclist is simply to ride more.
Some do's and don'ts for you:
Do ride more
Do make sure you have a good fit on your bike--a poor fit could cause injuries!
Do use clipless pedals--they help maximize your power
Do gradually increase how far you go (both time/mileage)
Don't fixateon speed right now, build up the endurance and "time in the saddle" (commonly referred to as T.I.T.S here on BT, so if someone says more "T.I.T.S" don't be offended!!)
Do buy a trainer if you are limited on the time (daylight) and location to be able to train.  Due to my times I can ride, I have done lots of trainer rides as part of my mileage accumulation.

You didn't start swimming and running by doing a 4000m set or 15 mile training run....while there is definitely some carry over from each to discipline to the other, each is also very individual so you have to train that discipline to get better at it.

Really? and to think my Local Bike shop made such a convincing sale!
Surely it can't make THAT much difference?

It makes quite a substantial difference -- you lose pulling power with the toe clips.  You can't really pull at all unless they are cinched so tightly that you can't get back out of them and you'll fall right down.  With clipless, the pedals are locked onto the cleats in your shoes, so you can pull pretty hard and not come off the pedal, and you can have a much more consistent application of force throughout your pedal stroke than is possible w/platforms and toe clips.

Are you possibly confusing clipless pedals w/platform pedals?