The obvious choice is an inkjet printer. The quality available these days is terrific.
You want to by a six- or seven-color printer. These give much better reproduction of images than do the standard four-color printers. These are often called "photo printers." A few will let you print directly from your memory card.
You will generally want to print on glossy paper made specifically for the brand of printer you have. Good ink control on a glossy surface is not easy, so you want to be sure that the properties of paper and printer match. Epson also offers a Heavyweight Matte paper that has a very different look, but is razor sharp, longer lasting, and less expensive.
Brands? I recommend sticking with Epson, HP, and Canon. They seem to consistently have good photo printers. The photos look like traditional prints. Models change fast in the printer market, so you will need to find what the current market is. Try to ascertain what the cost per print is. Sometimes it is cheaper to get a more expensive printer that has lower operating costs.
You can also get traditional chemical prints, just like those from film. Traditional chemical prints last longer than most ink-jet prints. You can check local photo labs for the service. However, far cheaper are the online labs. I have used Shutterfly, Ofoto, and Ezprints, and have been pleased with the results. I use Shutterfly the most, but I use Ezprints for long (several foot) panoramic prints. These services are quite affordable.
Also look at canvas wrap prints, available from places such as CanvasOnDemand. While more expensive that standard prints, the arrive ready for hanging. What you save on framing may pay for the extra cost of the canvas prints.
Walmart and Sam's Club offer traditional chemical prints in some of its stores at very attractive prices. You can drop off a CD or memory card and come back for prints. You can also upload to their web site and then pick them up at your local store. This service has not yet come to Winston-Salem, so I cannot comment on the quality. Reports I here have been favorable.
I find I rarely bother with 4x6 inch prints any more. I circulate "snapshots" by e-mail. Anything I want printed I generally print at 5x7 or 8x10. If I want a bunch of prints, I will use the online services to save my time.