I use Scarpa SL's (a B1 boot) with C1 crampons.
They are good enough, but, if you need to do a lot of toe work, then the boot will flex a little, loosing the ridgidity, which I personally feel would assist easier climbing.
B1 boot is a good starting point, but if you can progress to B2, then that might be the better option.
Ice axes, come in different lengths, so you might want to spend a bit of time getting one that suites you best.
My advice, take your friend with you if you can, if he's teaching you, then he should have a good idea what you will need and using any of the stores mentioned by Davey Mole, let the boot fitting specialist assist you in selecting your boots and crampons.
Not obligatory, but you might want to consider a helmet, just in case you do take a tumble.