TAKE THE PUPPY OUT WHEN . . . 
Immediately after he wakes up in the morning. 
After every meal and drink of water. 
After he wakes up from a nap. 
Before any extra freedom period. 
After extreme excitement and long play periods. 
The last thing at night. 

STAY ALERT 
between these times for signs that your puppy is "looking" to relieve himself 
(e.g., whining, acting restless, sniffing the floor, pacing around in circles). 
When you see him doing any/all of these things, try to distract his attention, then pick him up gently and rush him outside to his "toilet area." You may be going out 8 to 10 times the first few days; however, once the puppy settles into a routine, he should not have to go out more than 4 to 6 times a day, depending upon his age.

STICK TO A STRICT SCHEDULE. 
The more conscientious you are NOW, the more successful the training will be and the less "troubles" you will likely face down the road. It often takes some patience to make your puppy understand what you want him to do, but he will adjust to your schedule in time. Of course, there will likely be accidents; however, that's part of raising puppies. When your dog "makes a mistake" in the house, NEVER HIT OR ABUSE HIM PHYSICALLY. 
Correct him HUMANELY: The words "no" and "baaad dog" are the only corrections you need. 
How you SAY these words can convey your displeasure VERY effectively.

ALWAYS GO OUTSIDE WITH YOUR PUPPY 
during the training period. You want to see when and where he relieves himself, 
and your ENTHUSIASTIC PRAISE will encourage him. Once the puppy is completely housebroken, 
it should not be mandatory that you accompany him outdoors. 
IF, however, you live in the city and/or don't have an enclosed yard,
you MUST ALWAYS GO OUT WITH YOUR DOG: NEVER LET HIM ROAM FREE.

When the puppy does relieve himself after breakfast, he can have another supervised free period before being confined in his crate again until his next outing, when you will repeat these same steps. The length of supervised free periods depends upon a puppies age. Once yours can handle a 30- minute free period with no accidents, give him more freedom by increasing his free time to 45 minutes, and so on. Your goal is to increase his free periods gradually until he needs to be confined only while you are away from home. 

If the puppy's training regresses, it's back to "square-1": 
Start the training program FROM THE BEGINNING once more.

1.The dog is NEVER wrong or at fault when it comes to housetraining.
2.Anger or punishment for unwanted behavior is counterproductive.
3.Praise for correct behavior gets results.
4.Consistency in goals and training methods is crucial.
5.Patience is the key.

1. Housetraining mistakes are NEVER the fault of the dog.
He may poop and pee in the house as soon as your back is turned. 
This can sometimes happen even straight after I have taken him out walking and thought that he has done 'his business'."I think the key here is that you say you "thought" he has done his business. 
You must stay out with the dog until he goes, doing both pee and poop. Sometimes this takes a long time. 
Sometimes a good brisk walk will help. If you have stayed out a good 15-20 minutes and can't stay longer and have to bring the dog in, you must watch the dog while he is in the house, crate the dog, or keep the dog in some type of pen or enclosure. A young dog may have to go every half an hour or so. Control the dog's food and water so that he is fed and watered at certain times and then taken outdoors immediately. I have let him/her eat whenever they were hungry so if you have trouble with training you may want to put them on a shedule about 2-3 meals a day is sufficient for a puppy.

2. Anger or punishment for unwanted behavior is counterproductive.
Anger has no place in working with dogs, Anger will only make things worse. 
Tell yourself that you will work with the dog without anger, and that you will not allow yourself to get angry. 
Your anger can be controlled and your dog and you will both be happier. 
When something does make you angry, walk away and come back when you are calm. 
Decide that the next time the event happens you will deal with it without anger.

3. Praise for correct behavior gets results.
I think that rewarding good behavior has a much better effect than scolding or punishing bad behavior.
I think this is true for all training. Remember, your interaction with the dog is not only about housetraining. 
When you scold the dog for doing something that he is confused about you are just as likely to create another problem as you are likely to solve the housetraining problem.

The dog will not  be able to connect the accident that happened at 10:00 a.m. with the scolding he gets at 4:00 p.m. anyway.

When you see the dog beginning to look for a place to go, take him out and praise him.

4. Consistency in goals and training methods is crucial.
This is where consistency pays off. You may be trying too many things and the dog is confused.
Decide on the one thing you want from the dog and praise the dog when he does that. 
Decide if you want the dog to be trained to go outdoors OR indoors on papers. 
While some dogs learn to do both, most dogs will find it confusing to be expected to go inside sometimes and outside others. I myself reccomend going straight outside it can be to confusing and create bad habits later if you paper train. But if you really want the dog to inside rather then out then instead of paper training I suggest a Litterbox, It works much better and you don't have to worry about the dog "going" on every piece of paper left on the floor when he/she is older.

Work towards simple consistency. Let your goal be to train the dog to go outside and reward the dog with mountains of praise when he goes outside. Praise the dog for pooping outside and peeing outside every single time. This makes the time fun for the dog. Your dog wants to please you.

5. Patience is the key.
You have only had your dog for two months which is not a very long time in light of the fact that your dog could live another 10-15 years. The time you invest now in good training will pay off many times over in the years to come. 
Also, your dog is still very young. Be patient. We don't expect a two year old child to be potty trained and expecting a 2-3 month old puppy to be housetrained is asking too much.Every dog is different. Some may be housetrained at 3 months, others may take much longer. Also, keep in mind, when you are housetraining your dog you are also teaching the dog many other things about what your relationship will be. 
You are molding the dog's personality.

Your dog wants to please you. Give your dog numerous chances throughout the day to please you by going outside. 
Do everything you can to reduce the possibility for mistakes indoors by always watching your dog when he is inside or confining the dog until he is housetrained. For your carpets it is important to get something that will remove all traces of the odor to keep your pup from wanting to return to those spots. One good brand name is Nature's Miracle but Vinegar works well to.

I hope some of this helps. Take a deep breath, Relax. Focus on the good things going on with your pup. 
Your dog will be housetrained. Reward the dog for good behavior and make these formative months a happy period for both of you. Be consistent in what you expect and generous with your praise.
Most of all Be patient!