You've gotten some good technical advice on how to acheive the bokeh effect. But it's probably all gibberish to you, so the first thing I'm going to recommend is that you do buy yourself a book. Yup. A book.
Buying a better camera won't help you much if you don't learn the skills needed to operate it. It would be like buying a new motorcycle, but not knowing how to ride one - just because you can ride a bicycle (phone camera), doesn't mean you can ride a Harley (DSLR).
A book on the basics of photography, such as one of the Dummies books, will help you to improve the photos you take with any device, even your phone cam. For example: The Rule of Thirds, which is a photographer's rule of thumb stating that a photo tends to look most balanced if you divide your photo into thirds (horizontally and vertically) and your main subjects line up with one of the dividing lines. How do you do that? Well, in your iPhone camera app, there is a setting to turn on "gridlines". Turn them on and you'll see a tic-tac-toe board appear on the screen. When taking a photo of a person or vertical object, line the object up with one of the vertical lines. Line up horizons, ceilings, or other long horizontal lines with one of the horizontal guidelines. Bingo! You just learned the Rule of Thirds, and your photos will start to look more balanced.
Other nuggets of wisdom like the ROT can be found in almost any photography book, as well as more technical information regarding exposure (how much light strikes the sensor) and composition (how the stuff in your photo is arranged within the frame). Exposure and composition are the basics of photography - learn those skills, and you can apply them to any camera.
As for a specific camera recommendation, I'd recommend that you consider either a point-and-shoot (pocket) camera with manual controls, or what's called a "bridge camera", which is an automatic camera which looks kinda like an SLR, but has a non-removable lens. I recently bought a Canon SX50HS bridge cam, and I consider it a great beginner's camera. It's lightweight, has great image quality and some very nice advanced features, shoots terrific HD video, and has an unbeLIEVably long zoom. It retails for about $350. It's getting a little hard to find, because it's a discontinued model; the SX60 came out earlier this year. B&H Photo, Adorama, and Walmart all have them. I got mine from BestBuy, but I don't see it on their web site any more; you can check your local Best Buy to see if they still have any, but they'll probably only have the SX60 in stock.
Canon Powershot SX50 HS on B&H Photo web site
If you buy a new camera, budget for some accessories right off the bat: A padded camera bag for travel and storage; at least two 4gb or 8gb memory cards (many small cards are better than one big card, in case of loss or damage you don't lose all your photos); a spare battery (I get most of my spares from
StirnlinTek); a better carry strap than the one that comes with the camera; and maybe a wrist or hand strap. Also, a quality microfiber lens cleaning cloth is a must.
But get the book first.