The job of a photographer is to record the light that bounced off an object, nothing more nothing less. You don't need a lot of fancy equipment or a large collection of lenses to get good photos. I've seen beautiful work captured with a Pentax K1000 full manual camera, and I've seen junk captured on a Canon D5 Mark II. Just like owning a Porsche won't make you a race car driver, owning a high end camera won't mean you can get high end photographs.
First things first! You need a camera.
I recommend that if you are serious about getting good photos you start off with a camera that will allow you to have full manual control over the exposure, this will allow you to get certain artistic qualities from your shots that a point and shoot will not give you. So where does that leave us? When it comes to digital cameras, your choices are limited by your budget. The Sony's Alpha series, Nikon D3x00 series, and Canon's Rebel series offer manual exposure and they can be found used for $250~500 with a kit lens, battery, and memory card. If you are on a very tight budget I recommend that you take a good hard look at 35mm SLRs from Pentax, Minolta, Canon, or Nikon. If you want you can still get digital images by ordering a PictureDisc when you process your film. Some of Minolta's mid '90s model Maxxums sell for $30 and up, and a compatible lens can be had for between $30~150.
A "Normal" Lens
To get started, most photo instructors recommend that you get a lens that replicates the view of the naked eye. On a 35mm camera this will be a 45mm~54mm "Prime" lens. If you are considering one of the digital SLRs on the market (except for the Sony Alpha 900, Nikon Fx, and Canon 5D Mark "full frame"), look into a 35mm focal length lens. If there is only one lens you can get your hands on, get a "Normal" lens for your camera.
A Wide Angle Lens
A 28mm (17mm~19mm on most DSLRs) will let you get wide sweeping views of your surroundings. Architectural photographers love wide angle photography, my friend Mike Ryan actually likes the strange perspective given by getting too close to your subject with a wide angle lens.
A Telephoto Lens
Get up close and personal in portraits without getting uncomfortably close to your subject. An 80mm(55mm digital) or a 105mm (70mm on digital) lens is a nice lens to have in your bag. Many of the zoom lenses that are included with digital camera kits cover this range out of the box. A common zoom lens found on most beginner's camera's is a 28mm-80mm (18mm-55mm on digital)
A Strap.
'Nuff said. As long as it's comfortable and strong enough to hold your camera on your neck, you're okay. All camera manufacturers make straps that match the styling of their cameras. With my Film camera I prefer the thicker Minolta strap that has a bit of a velvet feel to it. It's also wide and doesn't dig into my neck. I have the Sony strap that came with my Alpha 230.
Film
Choosing the right kind of film can be a task. I recommend if you are starting out, to experiment with different options. There are Black and White print films, Color print films, Slide film, Infrared Film, Daylight Film, Tungsten light film. Only you can decide what look you want your photos to have, so buy different rolls, and try them out. In Black and White, I'm a big fan of Kodak T-Max 400 or BW400CN, and Ilford's HP5 400 ISO films. For color, I've liked the results from Fuji Superia 800 speed film and Kodak's SuperMax 800 ISO film.
Choosing a memory card is a lot easier on digital cameras than some make it seem. Only two questions to answer: What brand of camera do you have? and How many pictures do you want to be able to take? All entry level SLR cameras (Sony a230, Nikon D3x00, Canon Rebel) feature a slot for a Secure Digital card. Depending on the age of the camera you will want at minimum a 2 GB card. Sony's cameras offer two slots, one for SD cards and one for Sony's Memory Stick Pro duo cards.Higher end cameras (Nikon Fx, Sony a900, Canon D5) use Compact Flash cards which are physically bigger and are available in varying capacities. The higher the resolution (megapixels) of your camera, you will want to use a higher capacity card. (I currently have a 4GB SDHC card in my Alpha and I have two 8GB SDHC cards on standby for an upcoming trip)