Many companies make diapers specifically for newborns. I'd not recommend for or against investing much in this until baby arrives, unless you somehow magically know just what size baby you'll have. (I know, it can be estimated by sono, but it can also be waaay off)
Ours was 9 lbs + at birth and outgrew his newborn diapers in about two or three weeks. We had 5 all in ones (AIO's) from gro-via, about three or four more all in ones from another company (I'll add that in later, loved them!) and about four or five more covers that were exposed PUL style.
We were using Green Mountain diaper's orange prefolds for newborns at the time. I was lucky enough to find these on sale in our local consignment shop. They were about $6 for a dozen, used, and I got two sets. New they'd be $25 per dozen at Green Mountain. I really like the color coding (edges are colored), though it is very obvious they are way smaller than our next size up. I'll include a photo of the sizes in the inserts section of the diaper info. It was lovely how they fit for that time, though many newborn clothes are only meant to fit from 5-8 lbs. He was about 10.5 lbs by 2 weeks, so we were kinda stretching it keep diapering with the newborn AIO's.
Gro-via had a newborn all in one sale right before our son's arrival, so I ordered 5 of their discontinued prints at about $11 each. This was a great deal for these AIO's. These are also the only AIO that we've used, since I'm really into getting as many uses out of our shells as possible, and an AIO needs washing after just one use. These totally stayed away from his cord stump.
The image below shows a normal Gro-via hook & loop shell on our 3-day old. You can see under the diaper top that we've put it on very loosely and that his umbilical clamp is in place. We ended up leaving the clamp on until the whole stump fell off, so we really tried to either not let diapers touch it, or keep them loose in front. You may also be able to tell that there are scratches on his inner left thigh from the velcro. This is my main reason to go for buttons instead. I strongly prefer snaps. I just don't want to cause our LBD any discomfort, especially if it is avoidable!
Also, in the picture above, you can see that the pink cuff meets the flowery pattern without any snaps exposed. We have it on the tightest rise setting using all the snaps to make it short enough to fit our little guy. Here's a picture of another gro-via diaper on our newborn at 4-days old. The image is really dark because we kept the lighting low at first, but you can probably see the gap in the leg area and that the front of the diaper elastic on the inside leg is by his knee. We didn't quite have the hang of fitting diapers yet. :) This is where patience and persistence come in...
Since we had a newborn and were just adjusting to the high-maintenance lifestyle of a baby, it was pretty nice to have a diaper we just put on and took off - no inserts or adjustments needed for newborn size AIO's. If you don't mind shelling out the cash (pun intended!), or if you have access to consigned or hand-me-downs, these may be a good option for the first lil while (depending on size of baby). Or maybe you'll love All in Ones and just go with these for all your diapering - it's comparable to using disposables in that you just put it on and take it off. These cost more up front, but they are all you'd need.
I did not like the newborn Bummis and the "aplix." The (aplix) velcro was hard to keep away from little baby dude's skin and made little scratches on his super fresh tummy. They do a good job of fitting away from the cord stump, but I was otherwise unimpressed. I thought they were so cute and had high hopes, but would not use these again. I would consider trying their snapping style wraps.