I love smoking a pipe. I go through periods where I will have one or multiple bowls a day, and then periods where I just can't seem to find the time. I haven't smoked a bowl since Christmas time but I hope to change that this weekend. I don't smoke cigarettes, and it's been years since I've had a cigar, but I'm sure I will again at some point if one is offered to me.
As noted above a great way to get started is with a corncob pipe. They are cheap and smoke well, even after all these years I often prefer them to a briar - and am not alone in that opinion. Make sure to get a missouri meerschaum and not come cheap knock of crap.
I recommend the pipes magazine forums (do a google search) as there is a lot of helpful info.
Most folks start with what's referred to as "aromatic" tobacco. This is heavily scented to smell good for others, and offers little taste or satisfaction. These tobaccos also tend to burn hot and wet and can cause some tongue bite. Also do some reading on "English" blends, along with "Viginia", "Perique" and "Latakia" tobaccos.
With all tobaccos it's generally helpful to dry them a little, and make sure to draw from the pipe slowly, and infrequently. Really just trying to keep the tobacco smoldering, not burning. It's important to note that with pipe smoking you are not supposed to inhale. Just draw into your mouth, and possibly exhale through your nose if so inclined.
Pipe smoking takes some time to learn, and there is definitely a curve for most folks. I smoke in our office with an intake fan pulling air through a carbon scrubber and there is no scent. I am surprised to hear how many folks like the smell of my tobaccos - many have fond memories of their grandpa, or it reminds them of a campfire. I get lots of interested questions from folks as I often enjoy a pipe will walking my dog in the warmer seasons.
I'm going to disagree with all of this, except for the last bit. If correctly smoked, a pipe is much more gentle than cigarette or cigar smoking, as the temperature of the burning tobacco is lower. If your pipe is smoking hot and it is rough or it burns your mouth you're pulling on it too frequently or there's too much moisture.
Often, pipe tobaccco is extremely high quality, on par with or exceeding that which is used in many cigars. There certainly is cheap stuff that comes in a plastic pouch at a gas station (see above about "aromatic" tobaccos), but tinned tobacco is generally expensive, and of very high quality depending on your brand of choice.
I would steer clear of a super long stem if one is learning - they can build up moisture, leading to a hot burning pipe and tongue burn.
Smoking a well chosen cigar plug from a pipe is awesome, though.
I rotate through a few.
This one is from before WWII. Still smokes like a dream.
Peterson Early Morning Pipe or Capstan might be a nice place to start as they are mild.