I used to occasionally provide mechanical support for triathlons, and on the sprints especially I saw all kinds of bikes, from ones that people just dropped $5k on to ones they borrowed that their neighbors dug out of the basement. You can certainly get started with anything, and if you are reasonably fit there's a good chance you'll pass at least SOMEONE who spent more on their bike than you did.
But that said, you can get a very substantial and noticeable performance improvement over the bike you have fairly easily. The tires and a rigid fork are the biggest factors. But I agree with everyone else who recommended against spending the money on this bike, especially once you factor in the labor costs of having your LBS do it for you. Start training with what you have, and keep your eyes open for something better suited.
As far as saddles go, that's another issue entirely. Again, start with what you've got until you find out if and how it doesn't work for you. There's no one saddle or even category of saddles that optimizes performance or distance comfort or anything like that, and there are many variables. There's a good chance you'll need something narrower than what's in the picture you linked to. But again, you'll have to asses the needs of your own posterior as you find out what they are.
As for pedals, having some way of attaching your feet to the pedals really does give you some performance benefit. If you buy clipless pedals and shoes, they can perfectly easily be transferred to any other bike you get later. So that's one case where maybe it is worth it to buy them sooner rather than later since you'll be able to keep using them regardless. It can easily run you $100 for pedals and shoes, and there's also the complication of positioning the cleats. So another option is to start with toeclips (those cages with straps that you slip your feet into) on the flat pedals you have, which you can use with regular shoes, even your running shoes. Not to mention that in a sprint triathlon, not needing to change shoes in the transition might just save you more time than you'd gain from riding in good bike shoes for such a short distance.