What to expect At Pet Shelters:

Pet Shelters are your local Animal Care & Control, SPCA, Humane Society, they are a brick and mortar building that may or may not include foster homes. 

At most county run shelters, ie) Animal Care and Controls, you can walk in and view the available pets ready for adoption.  Some facilities function on a 1st come 1st serve basis, meaning who ever brings the paper work from the cage gets first dibs on that pet.   They then allow you to spend time with the pet, talk with a adoption counselor about the pet, they usually ask a bunch of questions to make sure this is the right pet for you.  They will ask you to fill out an application, show proof of home ownership or rental agreements allowing a pet and may not release the pet until all members of your family (including other pets presently in the home) have come in and visited the pet.  Most have test results on the pet showing temperament ratings and other pertinent information.  Some but not most require you fill out the adoption application first.  They may also require a separate adoption agreement.

Most shelters have a get acquainted room where you can visit with the pet uninterrupted, unless it is an exceptionally busy day.  Usually the adoption counselor  accompanies you.

A good shelter organization takes the time to discuss the many aspects of the pet, making sure this is the right pet for you.  They also share the information they have gather during the animals stay with them.  They may or may not have previous owner information.  Usually that information is limited to lost, stray, owner relinquished due to ...., returned to shelter because ....

Most shelters smell like pets.  The cages will probably have a mess or two, but should look like they are cleaned daily.

Most shelters have printed literature on various behavioral issues for you to take home.  Suggest you take one of everything.  Some have folders with relevant pet care information and pet care topics.  If multiple people apply for the same pet, they usually consider everyone.

What to expect At Pet Rescues:

Many rescues require you fill out an application before meeting the pet.  After they review your application they contact you with the time and place of their next community event so you can meet the pet.

Most Rescues do require a home visit.  They are quite concerned about their animals and do not jump at the first person.  This is a good thing, remember they rescued them from kill-shelters, nursed many back to heath, they have put much time and in most cases, much money into getting this pet ready for you.

The foster pet person should have lots of information on how the pet behaved in their home.  In addition, how the pet progressed, and any issues still needing work.  The foster person is usually the one who has spent the most time with the pet and either speaks to you directly or through the organization.

Like with the shelters, most Rescue pets are neutered and have a microchip.  Do ask if they don't offer the information.

Like Shelters, the Rescue folks should have information on IF the pet is suitable for home with children or other pets.  Again, if there is any question it is best to find another pet.

At Both the Shelter and Rescue: There are adoption fees for both. The amount varies depending on the organization and particular health issues of the pet.

Generally require: an adoption application and an adoption agreement.  They may have other paperwork as well.

Have you read Choosing a Pet?  It may give you some additional insight.  Have fun finding your new pet!