I hate to say it, but 3 weeks is still early days! Probably not what you want to hear, but I would suggest that you will have it sorted by about 6-8 weeks...
I've never been that keen on expressing, I found with my first that it just added to the misery (DC1 refused to latch on till day 17, so I did a lot of expressing up to that point!), so I would suggest that unless you really need someone else to be able to give a feed, you consider taking that out of the equation (I know some books say it is vitally important!).
As soon as you feel able, try to get out of the house for long walks, it will help the baby to go longer between feeds and will give you a bit of a break (I rarely slept in the day when DC1 was little, as he liked sleeping on the move...). You ideally want baby to be properly hungry when he feeds, as he'll be more efficient, and take more...
Once you get to 6 weeks or so, really try to work on not always b**bing to sleep, and also consider trying to ensure that you feed baby on waking, rather than when he is tired, that seems to be one of the most important things when trying to teach baby to settle, if they don't always associate milk with sleep...
Oh, and get some on demand tv! I spent most of the first 4 months on the sofa with a sleeping baby on me (or wandering round the local posh private estate on my long walks, with a nice cafe towards the end of the route!) and sky+ was essential!
Good luck!