Before I start: Past a certain point (where the camera does not limit the ability of the photographer to take the photos they want to), it is the photographer who makes a photograph good (with composition etc.), NOT the camera.
This question is one only you can answer completely, however, as always, guidance on where to begin can help. Below I have made a brief summary of the main points for each class.
A compact camera (do not confuse with a Compact System Camera, which are much the same as a DSLR) is generally easy to use with many automatic functions. It has a fixed lens (i.e. you cannot choose what range of focal lengths to have available), often with the ability to zoom from moderate wide angle to moderate telephoto. Without the ability to choose lenses it is often more difficult to achieve special effects.
Compact cameras generally also have much smaller sensors than DSLRs and so, as an unavoidable consequence, have more noise at high ISO and less ability to achieve narrow depth of field (often used to enhance artistic appeal).
Despite often being viewed as only for the casual happy snapper or not-at-all serious photographer, many compacts are now being designed for more advanced groups, for example Sony's RX100 or RX1.
- Small and light
- Easy to use
- generally cheapest
- less flexibility in regards to lenses
- less manual control
- less control of artistic effects
- complete Beginners
- anyone without an interest in the artistic side of photography
- anyone who wish to learn about composition etc. without the need to learn about DSLRs.
Without an deep understanding of manual functions and the more complex functions of a camera, a film SLR is probably not for you.
A DSLR is generally larger than a compact camera and has the ability to change lenses, allowing focal lengths of less than 10mm all the way to 800mm (and more if you are willing to spend A LOT of money). Most consumer grade DSLRs include automatic functions to make it easy to use the camera. These Cameras will be the most flexible and provide the best base for moving to more advanced (and expensive) cameras. Consumer grade DSLRs are comparable in cost to the upper tier of compact cameras, however tend not to be as easy to carry and transport. These cameras have better noise performance at high ISO than the lower tiers of compact cameras and have better control over artistic effects.
"Professional grade" DSLRs tend to be far more expensive and have less automatic features, instead requiring a better knowledge of the principles by which cameras work.
A Compact System Camera is similar to a DSLR, except does not have and optical viewfinder or mirror, but it can have many different lenses. They are generally smaller than DSLRs, with comparable image quality and flexibilty, however are easier to store and transport.
- greater flexibility
- more control of artistic effects
- greater room for improving photographic skills
- bigger and bulkier
- often more complex to use
- someone wishing to advance their photography skills
- someone who would like to have a large amount of artistic control over photographs
- someone who wishes to have a large amount of flexibility