As your puppy learns to control his bodily functions, it is inevitable that he (for reading ease, I will use "he" instead of "he/she") will have accidents in the house. (Catching him "in the act" is the best way for him to understand what is not acceptable.)
CLEAN-UP IS CRUCIAL
Many dog-owners use vinegar, but even though vinegar removes the urine odor from our noses, it won't from your puppy's nose. He will still be able to smell it, and he will pee where he smells pee. The best product to use to remove the odor completely is a natural enzyme cleaner, such as Nature's Miracle.
After your puppy is controlling himself in the restricted area, you can begin to expand access to the rest of the house. This doesn't mean that you can never bring your puppy into the living room, say, if you've been keeping him in the kitchen. Just supervise him at all times; he might have an accident. If you are paying close attention and catch him "thinking" about going, you've taken a big step in demonstrating that "in the house" (not just the kitchen) is off-limits. Gradually introduce other areas of the house. I guarantee that the first time he goes upstairs (or down in the basement) he will "go," no matter how well he's been doing on the main floor.
A note here about male puppies: when you take him out to pee, don't let him near a vertical object like a bush or a tree. Even though a very young puppy won't "lift" his leg initially, you'll want to avoid giving him the opportunity. Let him get used to peeing without anything around and he will be less likely to lift his leg to "mark" later. If he does try to mark in the house, correct him with a stern "NO" just as you would when you catch him going potty and take him outside. Another technique is to put some coins in a can and tape the opening shut. When the puppy is about to potty, drop the can near him (not AT him) so he is startled. Do this discreetly--the idea is to make him think it's the "house" that is startling him, not you. As soon as he is distracted from the act, rush him outside to finish the job. Remember to praise him when he's done!
Puppies should be brought out to potty 15 to 20 minutes after eating or drinking and EVERY TIME they come out of their crate, wake up from a nap, and after playtime. It is possible to train your puppy to potty on a schedule that works for you, instead of waiting for him to "tell you" when he has to go.
MORE ABOUT CRATE TRAINING
When your puppy is very young, keep the crate in his restricted area during the day with the door open. He will most likely start to go in it to take a nap. Periodically put him in his crate while you are at home, instead of just when you leave. This will give him confidence that going into his crate doesn't mean "My people are leaving!"
GETTING HIM INTO THE CRATE
The crate should be your puppy's "safe" place; don't put him in the crate for "punishment" or raise your voice when you put him in it. Many trainers recommend throwing treats into the crate, I've found that putting the puppy in the crate matter-of-factly from the very first moment causes the puppy to think that this is life as normal. Give a command such as "kennel up" every time you put him in his crate. Before you know it, he will run right in when he hears "kennel up!" (For safety, remove your puppy's collar whenever he is in his crate.)
Give your puppy some toys to occupy him while he's in his crate. Hard chew toys are best, like Nylabones or Kongs. You can put a soft bed in the crate, but if your puppy has a party and rips it apart, he should lose that privilege!
GETTING HIM OUT OF THE CRATE
Never let your puppy out of his crate when he is barking or whining. Wait until he is quiet. When he learns to "sit" on command, tell him to "SIT" before you open the door. (Remember, he is learning to sit each time he gets fed.) If he starts to get up from his sit, shut the door and say "NO." When he sits, praise him, then open the door again. He should not get up to exit until you release him with "OK."
Never let your puppy out of the crate as soon as he sees you in the morning, or when you come home. Let him settle down from several minutes to 10 or 15 minutes. Ignore him so he learns he must wait for you--this shows him that your are his leader. If he carries on in the crate, toss the coin-filled can near the crate to distract him. Or leave the room. It is important not to give any attention when he is misbehaving in his crate. In your puppy's eyes, getting yelled at is still getting the attention he craves.