Good photo's are very little to do with the camera and more to do with the person taking the pics.
You obviously have to accept certain limitations with a point & shoot over a more advanced camera but I would say that the least important thing about taking a decent picture is what you use to take it, and the most important is what you are taking a pic of, how you light it etc.
If you have a point & shoot camera you can still get good results but you need to understand what the camera is doing, what you might have to do to work around those limitations and how you might adapt your picture taking technique....
If you are spending enough money to warrant getting a DSLR though and you are interested in improving your photography, rather than just taking snaps then you might as well get one.
I can't see the pic you posted (just a red cross) so can't really comment on how you might go about achieving something similar.
If you know very little about cameras, what all the settings do, how to get different effects etc. then a few £ spent on some books will pay dividends.