It's great that he's starting to go outside
Keep at it and don't get discouraged (trust me there will be plenty of times! lol), your new puppy will be housetrained well in just a few weeks and you won't have to worry about that anymore.
Linds brought up a very good point to housetraining (and any training for that matter) is to attach a "command" word or phrase. Everytime you go out with your pooch and expect him to do his business, say the command a few times and very happily and enthusiasticly when he does it outside, kinda like goofy baby talk. Dogs love to please and he will associate the proper behavior with the command soon too. When all is done, all you'll have to do is tell him his command and he'll go! Both my dogs are trained this way and it works on every dog, even the difficult ones.
What's really important in house training ESPECIALLY with a very young puppy is that they have very small bladders and very little control over their bodies. They are learning their bodies and it's functions just like a human baby so they NEED your tolerance and patience during this training period. You cannot expect a puppy under 6 months of age to be 100% housetrained. It can happen and it's great when it does but you have to think for your dog's potty needs for the next 3-4 months, get used to it!
I cannot stress enough that taking a puppy outside in the proper area for him to go is very very important. As you've already noticed, they can be ready to go again in just minutes!!! Those are especially at the key times mentioned earlier: morning/long time before he went, and usually has a drink when they get up, after meals/full stomach pushing on bladder and digestive tract making stuff wanna come out!!! Go at least twice in those times!!! Do come back in between for a few minutes so he starts making the correlation of being outside.
Crate training is usually quite simple. The time you use it most is at night, other wise it's for corrections. When he's gone potty for the last time and you're going to bed, put him in the crate. Have a blanket and toy in there so he's comfy. If he cries you have to tough it out and let him until he stops. If it's really driving you nuts, after 1-2 hours take him out, go outside for a few minutes and put him back. Do not give him any attention other than to bring him out, talk to him gently and praise heavily if he goes potty. Then it's time to go back to bed and that's it.
In the morning, get ready to take him out before you take him out of the crate. It should be the first thing after you're up and you've gone. Take him directly outside and wait till he goes, praise praise praise when he does. Go back in, do your stuff whatever and take him out again in 20-30 min. Wait till he goes. After that he should be good for an hour or two!!!
If he messes up, that's when the crate is really nice. That is his punishment. Scold him for his booboo and put him in the cage tying another command to it like "go to your cage". Leave him in there 20 min and completely ignore him.
In time accidents will be less and less frequent and you will have a dog that pisses and shits on command and goes to his cage when naughty on command!