The first question is, how much are you willing to spend? There are many categories you can fall into depending on your budget.
Point and Shoot: These are cameras without interchangable lenses. The lens is fixed to the body of the camera. The idea behind them is to, as the name says "point and shoot" (without much thought). These will be the least expensive option, but also the most portable. These are not bad options if you are just trying to wrap your head around composition and don't want to spend a ton of money. Most people shoot with them on AUTO, but if you did go this route, I would still suggest that you learn how to shoot on MANUAL, because that is what you will need to know when you want to be serious about getting the correct exposure and to have sharp images by controlling the shutter speed. Point and shoot cameras have the smallest sensors, which means that the pixels are smaller and don't have as good as image quality as the interchangeable lens cameras. They also suffer in low-light performance, which means they will have more grain in the image as it gets darker. The sensor size is the more important aspect to consider than "megapixels". A 24mp point and shoot camera will not have the same image quality as a 24mp full-frame camera. People generally do not edit photos from these camera as they mostly shoot JPEG, which have the least amount of information to manipualte in the image as compared to RAW, that the APS-C and full frame cameras can shoot in.
APS-C: This is the middle tier of cameras, and the place I push most people to start when they want to learn and get serious about photography. APS-C refers to the sensor size and is commonly called a "crop sensor". They call it this, because when using the same lens as full-frame camera your image will be "cropped" in a little over 1.5x. This can be an advantage when shooting birds and other wildlife. This type of camera is an interchangable lens camera so you will have a wide variety of lens options. You can use APS-C lenses as well as full- frame lenses. It has a decent sensor size, but will not have quite the performance, especially low-light performance as a full-frame camera.That being said, people do use APS-C cameras for professional use. They are lighter and the bodies and lenses are less expensive than full-frame cameras. This type of camera ( and full-frame) will allow you to shoot in RAW, which means it contains all the information from your image instead of editing it in camera like with JPEG. You have control of how you want the photo to look by editing in a programe like "Lightroom". You can only edit a JPEG so far before it starts to get funky.
Full Frame: These are professional and high end cameras and come at a high-end price point. They will have the best image quality, low-light performance, and the best bokeh effect (smooth background blur). If you are using a camera like this, you will want to be shooting in RAW and editing your photos. You CANNOT use APS-C lenses on full-frame bodies because of the crop factor. You will have to use the more expensive full-frame lenses. I generally would not recommend someone dive into full-frame until they have learned how to control a camera and learn how to compose and edit images, but if you have a lot of disposable income there is nothing wrong with going straight to full-frame if you want the best equipment.
There are a few other options such as Micro 4/3rds and Medium format. Micro 4/3rds is pretty much dying out, and Medium format is too large to carry around, and very expensive.
Even though I gave you three styles of cameras that would be suitable for wildlife photography, keep in mind that there are tiers of cameras within those categories and will vary on features and price point.