Leave After Basic Training
With the exception of the Marine Corps and the Coast Guard, who authorize all recruits to take 10 days of leave, immediately following boot camp, one does not normally get their first military leave until they graduate technical school/AIT/A-School. After technical school/AIT/A-School, one is normally authorized to take 10 days of leave, if their first assignment is to a CONUS (stateside) base, and 15 days of leave if their first assignment is to an overseas base (Note: For extremely long schools, longer leaves may be authorized upon graduation). The 10/15 rule is usually automatic, and contained in the written orders, and apply even if it results in the recruit going into "the hole" on their leave balance. (Of course the person doesn't have to take all, or any of the leave-time authorized).
Christmas Exodus. During the two weeks around Christmas time, the Army pretty much shuts down basic training, and AIT schools. The Air Force and Navy do not shut down basic training, but do shut down many of their job schools (Tech Schools and A Schools). This period is known as "Christmas Exodus." Recruits are usually allow to go home on leave at this time, if they want, even if it results in going "in the hole" on their leave balance. Recruits who choose not to take leave at this time, are normally assigned to do details (answer phones, cut the grass, etc), because most of the instructors/drill sergeants will be away on leave, and classes are not conducted during this time. Usually (but not always) if a recruit takes leave during Christmas Exodus, that results in a negative leave balance, and they won't be authorized to take leave after school graduation (Note: For long schools, individuals may have had time to "save up" some more leave. They would normally be allowed to take that leave, after graduation, if they want).
In most cases, the cost of travel is at the member's expense, while on leave. However, in the cases of Emergency Leave, while assigned/deployed overseas, or deployed at sea (Navy/Marines), the military will arrange free transportation back to the States. Once the individual arrives at the "port of entry," in the States, the cost of travel to their leave area is up to them. Once the leave is finished, the military will also arrange for free transportation from the "port" back to the overseas/sea duty assignment.
Passes. A "pass" is non-chargeable "time-off." During a military member's normal off-duty time, they are automatically considered to be on a "Regular Pass." In the old days of military service, a military member had to have permission to be "off-duty," and to leave the base. To even go off base, they would have to have a written pass from their commander and/or first sergeant. Today, the "regular pass" is a person's military ID card. With a few exceptions (such as basic training, or phase restrictions in technical school), a military person can leave the base when off-duty without special permission.
Another type of pass is a "special pass." An example would be the 3-day pass. These are special passes issued by the commander, first sergeant, or (sometimes) supervisor for "time off," often given as a reward for superior performance. Usually, a special pass cannot be used "back-to-back" with leave, and cannot (in most cases) be used in conjunction with a weekend or other scheduled off-duty time.
Permissive Temporary Duty Assignment (PTDY). Sometimes a military member wants to attend a conference or class or function, that the military won't pay for, but which benefits the individual professionally (which thereby benefits the military). In such cases, the commander can authorize a Permissive TDY. Members on Permissive TDY do not receive any travel pay, or re-imbursement (like they would for an official TDY), but the isn't charged against their leave. Examples of permissive TDYs would be the Hometown Recruiter's Assistance Program, or a computer specialist attending a Microsoft certified computer course.
Hometown Recruiter Assistance Program
With the approval of the recruiter and the school commander, recruits may be authorized a "permissive TDY" to return to their home town and assist the recruiter (for up to 20 days, depending on the service). A "permissive TDY" means the military won't pay the travel costs, but the time is not counted against leave. Recruiter Assistance Duty can often be combined with leave, so the member spends some of their time at home working with the recruiter, and some of their time at home, on leave. The amount of leave that can be combined with the Recruiter Assistance Program, and the length of the authorized Permissive TDY varies from service to service.