DSLRs are what you might call ‘proper’ cameras because you get easy access to all of their controls which makes them quick and easy to use. They also have the widest range of lenses and other accessories. Generally they perform better too e.g. faster autofocus (important for animals, children and sports) and longer battery life. For this reason we always tend to recommend people buy an SLR if they are getting into photography.
People can be put off by the size of an SLR, which is understandable, however Canon have the 100D which is the smallest SLR available. They also have a 40mm lenswhich is really slim so would make a good smallish combo for walking around. This could be combined with the kit zoom. The Canon 650D and Nikon D3200 are also good small starter SLRs, although bigger than the 100D.
Good non-SLRs are the Sony NEX6 and Fujifilm X-E1. Both have the same size sensor as an SLR so similar image quality, and they are smaller, but you lose out slightly in terms of performance and handling. Both have interchangeable lenses. Either would be a good choice. You have a trade off between size and performance, and in our experience the trade off in size is not worth it for the loss in performance and handling.
There are many other good options including the Canon G15 or whatever its current iteration is. It has slightly lower image quality and you can’t change the lenses. It’s smaller again than an SLR, with the same aforementioned handling and performance compromises.
So, with it’s superior handling, decent autofocus, excellent image quality, and huge range of lenses and accessories, the 100D combined with the 40mm lens could be the way to go if you want a small, proper camera!