A lengthy and detailed article about halachic issues with pets can be found here. http://www.daat.ac.il/DAAT/english/halacha/jachter_1.htm
As the article points out, besides for the fact that most of the Gemara’s stringencies regarding dogs tend to hinge on the dog being “evil”, there is at least one instance, in Bava Kama, where the problem is only with security. It says that if you chain your dog so that he won’t damage property or frighten people, it’s completely permissible. Regarding other pets, the article cites two sugyas that have examples of children playing with birds and locusts. So apparently, having pets for recreational purposes (as opposed to service dogs) is fine.