This How-To is for individuals who have already completed their License to Carry course and shooting proficiency, and have received their LTC-100 or LTC-101 document. If you have't done these things already, you're skipping too far ahead. When you have the LTC-100 (in person) or LTC-101 (online) from your LTC instructor, at that point you're ready for these steps.
The application itself shouldn't take much time, assuming you have all of your required documents. Time delays may occur if there is a problem with your paperwork, your background check, or your fingerprinting. If you have any criminal record, you may be asked by RSD to submit additional paperwork (court documents, for example). Don't get discouraged, the processing time by the state takes as long as it takes.