You can technically make your own essential oils, but I would advise buying them from suppliers, especially when first getting into aromatherapy. Unless you have acres and acres of each particular flower/tree/fruit that you want to harvest (and there are hundreds of oils used in aromatherapy) you won't get much oil. For example, it takes about 7 and a half tonnes of the Melissa flower to make one pound of the oil, not to mention you have to make sure the crops don't get eaten by pests and hurt my nature and also you will have some years where harvests just suck. I haven't read about creating your own carrier oils, but I imagine the process would be very painstakingly similar.
The general guideline is to never use aromatherapy on pets. Essential oils are so concentrated that they are used in very limited amount on humans (a couple of drops when applied topically), and the reason we can stand them is because our liver has the capability to process the oils out. Pets are generally much, much smaller and don't have the capability. On amazon.com, I saw a review for an aromatherapy book that had a section for usage on animals. A reviewer complained that after using the remedy on her cat, the animal died. For humans, most aromatherapy oils can be used, but they should never be ingested by someone who has not run the idea by their doctor. Eucalyptus, for example, is great when used diluted on the skin, but toxic when taken internally.
You will generally find essential oils and carrier oils in most health food kind of stores. However, I would refrain from buying from these places as their oils are not always 100% pure- they can be diluted with carrier oils and sold as pure, or in case of expensive oils they can be diluted by a cheaper essential oil which smells similar. There are many good retailers online, so check them out. Also, I find hydrolats are easier to find online as well, since they're not very popular with people who are not familiar with aromatherapy.