The biggest thing you have to remember is that the camera doesn't take the photos. The photographer takes the photos. The camera is just the tool that the photographer uses. A DSLR is a more manual camera - meaning, it relies more on the photographer than a point-and-shoot. You could very well get better images if you use the DSLR correctly. But, they could actually also come out worse. My suggestion is to scour the internet on basic photography techniques and learn how to best take photographs. That will help you way more than purchasing a fancy camera.
All that being said, I personally am a fan of the introductory level Nikon DSLRs. Such as the D3300 or D5500 - they sell bundles with two lenses, carrying case, etc. They are great starter cameras, and if you get really into the photography hobby, you won't have to replace the camera (just the lenses). I used a D40 many years ago, which is now discontinued, but it was a great starter for me before I got really into it.