Sympathies - I bf my DS until he was over 2 so expected no problem with newborn DD, how wrong have I been! At first I got the latch wrong, as a newborn is very different to a toddler who knows what they are doing. However it takes 2 to tango and she doesn't open her mouth very wide... so I suspect things are only really going to improve once she grows more. Other possible causes can be tongue tie (as suggested already) or high palate. However as my DH said, my nipples also needed to 'toughen up' again, they'd reverted to being soft after 9 months of resting.
What the MW suggested to me at the start (when I was around day 6?) was to express from the sore side and rest it. This means less rest for you though - your partner can feed your DS an expressed feed but you'll need to be expressing to keep your supply up. You don't need to do this for long - perhaps 12 hours might help you get through the worst, or 24hrs. You can cup feed the expressed milk, if you want to avoid using a teat. Do watch your breasts though in case you get a blocked duct (will be hard and red) - this can lead to mastitis.
In the meantime to help your nipples heal, ask MW if they can prescribe you Jelonet - it's basically a vaseline-impregnated gauze, but it is quite soothing and has helped me heal up.
Ibuprofen might also help, if you have any idea when your DS is going to feed you can take some around 30mins before.
But probably the main thing is to work on the latch. I'm still not there, it's getting better but some feeds are better than others, depending on how frantic DD is for the milk... tonight I'm back to expressing.
Good luck!