Make sure the trainers you get are actually running trainers, not x-trainers or footy types. They will not allow anything but. If you can stretch your budget to getting yourself a pair of silver shadows, then get yourself a pair of them too, for inspections, as you will have to use your issued ones at some point.
As for ironing boards, yes, there are plenty, but not enough to go around everyone and if you rely on them you'll find that you may have to wait your turn, rather than be able to iron exactly when you want to. I'd get your own if i were you.
Most things are available from the NAAFI, but it is more expensive, so try and stock up on what you can for cheaper - i found that 'Flash' wipes were a godsend for block cleaning - stock up on them !
As for swimming caps, you won't need one and you don't need trunks necessarily - just normal swim shorts, plain in colour and not down to the knee, like bermudas.