Puppies/Kittens < 6 weeks of age: these animals are supposed to still be with their mothers, so if some misfortune led them to need a home, they will also require some proper “weaning†to be done (both behaviorally and physically). These animals may need fed milk replacer if they are very young; or canned food. They will require transition to eat dry kibble. They urinate and defecate a LOT and cannot yet be potty trained, so they require a lot of clean up. They need to eat at least 4 times daily; often more (depends on age). There is great personal reward in bringing up an animal from this young age. It is a lot of hard work, but there is also a strong bond that is formed through this process. Some behavioral problems may emerge due to lack of proper peer socialization so we humans need to emulate “moms†and “siblings†in our interactions. Do not allow for persistent suckling behavior, mouthing, or attention seeking. Moms push their babies away when they are fed and when mom needs a break – we humans must do the same. Siblings cry when they are bitten too hard, vie for attention, hog resources, etc. This teaches the puppy/kitten to cope and learn to not be “needyâ€.
Kittens 6 weeks to 5 months of age: they are now weaned and able to eat canned and dry kitten food. They are ready for a new home and are growing rapidly. Take lots of pictures! At first they sleep a lot and play hard in bursts, later on they just play a lot. They are open to learning and socialization, and are eager. Be sure to use this time for training them!
They can learn “house rules†quickly and develop some tricks as well. This is a great time to adopt, but remember their behavior can still be molded, so plan on spending time doing that. Kittens do not train themselves to be good cats – that’s up to you!
Puppies 6 weeks to 8 weeks of age: they still supposed to be with their 
mom and litter-mates at this age. Pups are learning core socialization skills and also coming through a “fear period†so re-homing at this age is not ideal, though frequently done. The reason some re-home at this age is because they are weaned off mom’s milk, however there is more to development than the physical weaning. If you get a puppy at this age be sure to focus on its socialization.
Puppies 8 weeks to 20 weeks of age: these pups are in prime learning mode and able to adapt to new environments, habituate to new situations, and learn new things about their world. This is the usual age when pups are re-homed and a great time to get a puppy. You will be responsible for how this puppy grows up and what it will be like for the rest of its life – that’s a big duty. It’s very fun to get a puppy at this age – they are cute, growing, and moldable. You do not know what they will be like as adults, but you have some important input into that development. So both the pros and cons for adopting at this age are the same: training. There are manyresources available to help with this process – use them, have fun, and take lots of pictures!
Dogs 5 months to 1.5 years of age: young dogs are still highly trainable, however potentially developing bad habits if their training and socialization have been lax to this point. Adopting a dog at 
this age means you will have your work cut out for you. They may have some core behaviors that are beginning to develop in the wrong direction which require correction. That may be the reason for re-homing them so do yourself a favor and find out the details. These dogs also need a lot of exercise at this age – plan on daily activity. The benefit to adopting at this age is that many of the dogs are already house-trained and you know more what they will look like when they are fully grown. You can avoid some puppy visits to the vet, but may still have to spay/neuter your new dog. If you have a busy work schedule and need a dog that can be home alone for 6-8 hours per day and only eats twice daily, this is a great age to adopt, just plan on doing your “work†with the dog in the AM’s, PM’s and weekends.
Cats 5 months to 1.5 yrs of age: juvenile cats have learned their core behaviors and think they can take on the world. Their behavior is more difficult to modify at this point, but it can still be done. If they have not been socialized by now, dominance can become an issue. If they are not scratching post and litter box trained by now they will have problems staying in their homes. This may be the reason they are being rehomed, so ask the right questions and try to see where they are living if at all possible. It is also possible that re-homing is being done now due to cost factors – if the kitten has not been spayed/neutered yet and vaccines cost more than the former owner planned for, you may be adopting a great cat simply because of someone else’s poor planning.
Cats/dogs 1.5 to 8 yrs of age: these adults have learned their core behaviors and are quite set in their ways. They can certainly be taught new house rules and tricks, but it takes more time and patience. Hopefully they are fully house-trained and socialized well. What you see is what you get, so take time deciding if this pet is the right one for you. The real benefit of adopting at this age is that you do not have to do the fundamental training. You can find a really well-mannered pet. Their personality is now predictable and you no longer wonder what they might grow up to become, how big they will get, etc. The down side is that some behaviors come out after they have been home with you such as dominance in cats and phobias in dogs – you can’t always determine these prior to adoption. Often these pets are being rehomed either due to the former owner moving away, or a behavior problem the owner can no longer cope with. No one wants to admit to the behavior problem(s), so ask a lot of questions!
Cats/Dogs >8 yrs of age: these seniors are not often in need of re-homing, but it does come up on occasion. Some people prefer to adopt a senior pet – they require less attention, exercise less, and are often quite content just “hanging outâ€. If they had a good home prior, they will make a great pet. Again, their behavior is pretty predictable. They are often being re-homed because of medical costs or the owner had to move away. Sometimes people discover their pet will need treatment for a long-term illness and decide they are not up for that financially, sometimes it’s just living circumstances. These former owners are often looking for a great home for their pet and will usually disclose their full history so they can be sure the pet won’t need re-homed a second time. Senior pets are also a great idea for people who understand that they, themselves, are in a short-term situation. They can adopt a pet knowing it will not live potentially another 20 years. These pets are great adoptions for older empty-nesters, too. Be ready to deal with the end-of-life issues that come along and be prepared financially to help these pets maintain good quality of life in their later years.
