Take your pet to a veterinarian in order to get it properly checked up. Anyone interested in adopting your pet will definitely want to know a bit about its medical history. It is best to be prepared for such a demand instead of making stuff up.
If your pet has some behavioural issues such as barking unnecessarily or littering at places where it is not supposed to, make sure you take care of this things before you put it up for adoption. Even though you loved it despite its flaws, do not expect others to show the same amount of unconditional compassion. In the worst case scenario, your pet may get adopted by a person who has absolutely no tolerance for behavioural issues and consequently get it euthanized.
Once you have gotten your pet ready for adoption, ask your friend, neighbours, colleagues and vet if they know anyone who may be interested in adopting your pet. If they give you a lead, go meet that person and take your pet along if possible.
In case none of your contacts knows anyone who may be interested in giving your pet a home, take a cute innocent picture of it and upload it to your computer. Start advertising it on the internet. You can put up an ad on social networking websites such as Facebook or Twitter, or you can simply place an ad on Petfinder, Craigslist or Kijiji. Print pictures of your pet using a colour printer, write your contact information on it and paste it around your neighbourhood.
If your advertisements fail to get you a response, take your pet to a local animal shelter. Ensure that they have a no-kill policy, so that your pet does not end up getting euthanized if it is not adopted within a given period due to space limitations.