Assuming you fulfilled the requirement of getting married to your U.S. citizen fiance within 90 days of having entered the U.S. on a K-1 visa, you are correct that it will be up to you to take the next step and apply for U.S. lawful residence.
We're assuming, of course, that a green card is actually what you want. Some people use a fiance visa just to get married in the U.S., and then leave afterwards.
The technical term for applying for lawful U.S. residence from within the U.S. is "adjustment of status." In brief, that means submitting various forms and documents to U.S. Citizenship and Immigration Services (USCIS), having fingerprints taken, and attending an interview at a USCIS Field Office.
This article will go over the basic procedures for the adjustment of status application.
Be aware, however, that the "green card" you receive will likely not be a permanent one. As someone who entered on a fiance visa, and only just got married, you're unlikely to have reached your two-year wedding anniversary by the time USCIS approves you for U.S. residence. By law, you will thus receive what's called "conditional residence," which will expire in two years unless you file an I-751 application to renew it.
Here are the USCIS forms (downloadable from USCIS's website) and documents that you'll need to fill out or otherwise prepare (which in most cases means making copies of them and supplying word-for-word translations if they're not in English):
After compiling all of this, make a complete copy for your own records and send it with your check, money order, or credit card form to the address indicated on the I-485 page of the USCIS website.
If this seems like too much paperwork to deal with on your own, of if complications are present in your case (such as a recent arrest or loss of household income), consult an attorney.
After you've mailed everything in, USCIS will send you a receipt notice, containing instructions for how to check its progress through the agency's processing. Be warned, long waits are common.
At some point, you'll receive notice to attend a biometrics (fingerprinting) appointment. Later, you'll receive an interview appointment notice. Both the immigrant and the U.S. citizen must attend this. It will be held at a USCIS office in your state.
If you change addresses, be sure to let USCIS know.
Prepare to bring with you a photo identity document, originals of all the copies you submitted, and any official documents showing changes to the material in your application (such as a legal name change). Also bring documents showing the your marriage is bona fide (real), not a fraud to get the immigrant a green card.
Arrive well ahead of your appointment time, because you'll need to get through a security checkpoint (it's a federal building). Also, if you're late, you might have to return on another day.
The interview itself normally takes around 20 minutes. A USCIS officer will swear you in, ask questions to make sure the information on the application is correct and that the immigrant isn't inadmissible, check the results of the immigrant's fingerprint report (which is run through FBI and other databases), check the medical exam report, and ask questions about your marriage.
Proving a bona fide (real) marriage is one of the most important parts of the interview. The officer will ask questions like, "How did you meet?" "How many people attended your wedding?" and "Does either of you set an alarm clock in the morning, and for what time?" If one or two of your answers don't match perfectly, that may be understandable.
But if it appears that the two of you don't really know enough about each other to possibly be sharing a life, you might be taken to separate rooms for a longer interview. An officer will ask each of you identical questions and then compare the answers. This could result in a denial of the green card.
If all goes well, your residence should be approved on the interview day or soon after. Your green card will come by mail some weeks later.