Painful as this feels at the moment, it is normal to be engorged in the very early days, and normal for the baby to struggle with the size/shape/hardness. In my experience, it is also normal for it to be very hard to get a perfect latch when baby is small and floppy, and boob is enormous and hard, so don't beat yourself up if the latch isn't perfect and you're a bit sore.
Things to try to help:
Hand express a bit before a feed to take some of the milk out and make it less hard. Or use a pump, just don't take too much milk off or you'll tell your boob to up production even more!
Warm bath/shower and massage boob.
Disposable nappy with warm water in it as a warm compress before a feed. The warmth will probably make you leak, but the nappy will soak it up :)
Put your fingers either side of your nipple and firmly press back into the boob to move the milk back into the breast from the areola. Latch baby quick before milk comes back down.
Feed a bit less often from the sore side if you need a break from the pain. Don't stop using that breast completely, but you won't mess up your supply or make wonky boobs just by using the more comfortable side a bit more in the early days.
Hang in there. My DC3 is 13 days old and we're well through the main engorgement phase. From memory (it was only last week!), I think the engorgement was worst for me on days 4 to 7, having had my milk come in on day 3 (counting birthday as day 1!). It really will get better quickly if you can keep feeding regularly.
I hope it gets better for you soon and your nipples survive the dodgy-engorgement-latch. Lansinoh will really help them - even though it's very expensive, in my experience you'll only need one tube and it'll last long enough even if you really slather it on after each feed. Don't hold back with it because you don't want to run out - you won't need it for long enough for the tube to run out :)