Why?  A stroller is one of the most important pieces of gear to have as you and your apprentice(s) venture out into the world, and one of the most complicated.  It will be useful from day 1 until about 3-4 years old when your kids can walk a decent amount of time on their own and must transform several times (or you have to buy several different ones) to accommodate their development and size changes.  But the real bottom line for me is that the Stroller saves you from carrying the kid and all of the support gear for the first several years and frees you up to move more easily on your journeys.

Options:  With all the different types of strollers most break down into a few basic categories: 

1. Normal size strollers: Pretty much what it says and the most common
2. Jogging strollers: Like a Normal sized stroller but built for speed and stability
3. Multi-kid strollers: These can be specialty models, or simply add-on devices
4. Car seat strollers: A frame with wheels that is designed to hold a car seat
5. Umbrella strollers: Light weight and smaller framed, for somewhat older kids

Each of these different types of strollers have a set of strengths and weaknesses that should be considered when purchasing a stroller, and as this is one of the pieces of gear that your entire family will interact with it is important to pick the one that will work best for everyone.

Considerations:

1. Safety:  Obviously the biggest concern is getting a stroller that is safe, so checking for recalls is a good first step.  Also making sure that the restraint system is easy to adjust and secures the kid well into the chair is important.
2. Ease of use: This includes maneuverability, handle height, foot clearance for under the stroller while walking, easy to close/open quickly (with one hand?), storage below stroller for diaper bag/shopping bags, storage at handle area for misc items, breaks that hold and are easy to turn on/off.
3. Comfort: It’s also important to look at the positions that the kid can use the stroller in, for example reclining as the fall asleep or upright when awake.  Having a good foot rest and snack tray will also come in to play later on.  You also want to make sure that the stroller has an adjustable canopy to protect your kid from the sun/rain.
4. Cleaning: This is a big one for me.  These things get pretty nasty and having a stroller with fabrics that are easy to wipe down and the ability to disassemble for washing is very important. 
5. Cost: oh yeah, that.  You can buy a stroller at just about any thrift store that is still in pretty good shape for $20 or pay close to $1,000+ for any number of new “fancy” designs, but at the end of the day it’s what you can afford and making sure that it fits your needs that is most important as they all really do the same thing.