Before everyone jumps on you, I should warn you that these kinds of questions don't appear to fare well here. The reason for that is because you've already made a critical error in assuming all Japanese people are the same, and there's some magic formula for how to meet any Japanese person under any circumstance.
But, having said that, GENERALLY speaking, its the same as any other part of the world. If you meet someone who takes a liking to you, kindly ask if they'd care to meet again and follow up politely. Over time the person may become your friend. Just keep in mind that the Japanese frequently (especially over drinks) exchange contact information and business cards out of politeness, but really never intend to contact the other person. If someone gives you an e-mail or phone number and it seems as though despite this fact, they do not wish to meet up again, don't get offended. In actual fact you may have made the mistake in assuming they wanted to meet up again in the first place, its a cultural difference between the West and Japan. In the West people exchange contact information with the sole intention of contacting one another again. In Japan its a bit of a formality which carries less social onus.
Furthremore, making good friends who will as you say invite you back to their homes and travel together and whatnot, well.. generally the home is not a place for entertaining, for receiving guests too frequently. The Japanese home is a private place for close friends and family mostly I'd say.. generally socialization, entertaining and whatnot is carried on in any of the limitless numer of bars, izakayas, restaurants, etc., where one can drink, relax and be a little noisy. This is where your new friend will most likely wish to meet until they get to know you better. This is also for the purpose of noise reduction as Japanese homes (especially apartments) are densely situated.
So yeah, try not to assume there is a secret part of Japanese society out there that loves foreigners and just can't wait to bring one home, and all you have to do is find the secret handshake. It doesn't exist. Making friends in Japan is the same as in any other place..you have to be polite and patient. There's no magic formula and even if you find someone cool, from my experience the Japanese are very busy and sort of 'slot you in' to their schedules where they can. Don't get frustrated if they're always busy and can only meet for a couple of hours a week.
If you want true cultural exchange try finding some local programs or groups at your ward center, or keep an eye out in the Metropolis magazine for individuals wanting language exchange and also a calendar of events if you're located in an urban center.