Anyone who has had an eight week old puppy knows that the first couple of months can be challenging. Usually you get a little separation anxiety and lots of learning the new rules, finding things chewed that shouldn't have been, and probably spend a lot of time cleaning up messes on the carpet as they get things figured out.
It's a little different with a four week old puppy (or younger, though I never recommend taking a pup younger than four weeks away from momma unless you absolutely have to!)
The learning curve for a four week old pup is different... they're at the stage where they're learning social cues from their mom and any other dogs around them, so they absorb that like crazy. Also at this stage is when they start figuring out that using the bathroom in their "den" isn't as happy as they thought it was a week ago, so if you're willing to put up with puppy pads, they learn quick that that is the acceptable place to do their business. Just make sure you have one nearby wherever they're hanging out.
Cause, see, they can't quite grasp the concept of the other room yet. There is "here" and "not here." If you're "here" that's great, if you're "not here" they have no clue where you are, and depending on the pup may set up a howl so that you can find them. At this stage they're just barely starting to learn that they could conceivably go look for you, so they might wander around crying hoping you hear them, but it takes a little time before they really figure it out.
Separation anxiety is huge around this age. They're still more likely to be sleeping in a pile with their siblings than finding someplace to flake out in comfort by themselves, and losing that comfort can be very upsetting. A lot of people (even me, at times, I admit) get frustrated when a puppy will not show any signs whatsoever of this anxiety until some gawd awful hour of the morning, but when you think about it it makes perfect sense. During the day, they're getting input, and their little brains are soaking it all up and learning. Presumably at gawdawful in the morning, you're asleep, the room is dark, and everything is quiet, so the distractions just aren't there. They go looking for the comfort they've always gotten from their siblings and mother, and don't find it, so they cry until someone comes to tell them it's all right.
Diet can be a challenge too. I've dealt with it enough times that it's old hat, but a lot of people wouldn't realize that at this age, they're not very good at drinking straight liquid, so you have to make sure they get their liquids in a way they can handle... like mixing milk replacer into canned puppy food.
The puppy food is another thing that can make a big difference. If you absolutely can not lay your hands on puppy formula canned food (it's getting harder and harder to find, since most people don't get pups till they're big enough for dry or at least semi-soft food, which is less messy and easier to deal with) then you can use adult canned dog food... but puppy style is so much better. It has a ton more calcium in it, and the Pedigree we usually use to supplement our pups as mom starts to wean them or gets a little overloaded as they grow (Pedigree did not pay me, does not know I exist, and could care less even if they did) works really well for mixing with milk replacer. It's very fine ground, but doesn't turn to absolute slop the second you add liquid, so it's easier for them to eat than something with tougher chunks in it, or that just melts into a puddle of slightly thicker goo.
A pup at this age takes a lot of time, as well. They have four settings: eat, sleep, play, and potty. All of these settings are all important when they're going on, and they are extremely urgent as far as the pup is concerned. A pup can figure out how to entertain itself in play mode, and they'll sleep just about anywhere, but you've gotta be ready for those to swap to either of the others, because if you're not, you'll be told, at volume, or spend time on your knees scrubbing the carpet. (I love puppy pads, I really do, there are three deployed in the house right now, living room, in the big kennel in the bedroom, and in front of the small kennel in the office where the pup stays with me while I'm working until DM gets home and is ready to take over puppy duties.)
The method of learning at this age is different too... positive reinforcement sticks ever so much better than negative. When something bad happens to a pup at this age, it just falls out of the clear blue sky, seems like. When something good happens, the association gets made ever so much more quickly. Not saying that you can't stop them from doing something you don't want them to, just that starting them on learning the word "no" and simply moving them away from whatever it is they were doing, and offering something you'd rather they do, seems to be ever so much more effective.
But you have to catch them right then. They can't even make the association that they will in another two weeks, of the experience of having their nose rubbed in a mess being linked with the fact that they made the mess in a spot they shouldn't. If you don't catch them actually in the act, you're pissing in the wind, confusing the puppy, and aggravating yourself, because they have no clue what the fuck you're talking about.
On the other hand, this is one of my favorite stages of development. They're learning so much every day that it's amazing to watch them. Their coordination improves nearly hourly, and they throw themselves into everything with full enthusiasm. They'll knock themselves over shaking a toy, and then you look away for a second and they've executed a sneaky puppy climbing mission, scaled the fridge, and set to trying to tear apart the cereal boxes. Luckily, they don't have the teeth or the coordination and strength to do any major damage too quickly at this age, and they seem to lose a few of the ninja skills as they grow and become less bouncy so that falling off of things actually kind of hurts.
(Yes, for those of you who have had pups at eight weeks and marveled at their ninja sneaking skills, they had more, believe me.)
Raising a pup from four weeks is not something you do lightly, but sometimes it's worth it.