I have never been able to run even a quarter of a mile straight. Even now, I have to stop and catch my breath. I'd like to take up running. What are the best ways to ease into it?
Best way to start is a couch to 5k program. Teaches you to increase your mileage a little at a time over nine weeks. It is a walk run program. You start out walking more than running, but eventually you do not walk any.
There a lot of training programs, some are built for your phone.
Just remember that you do not need to be running 5 minute miles. Slow and steady wins the race to completing the program. Speed comes with training or you are a freak and born with it!
Many runners will tell you they started at the same point you are now. Building a habit for running is, in my opinion, as important as the running itself. Every day you run a little farther. You will find many little victories along the way. The first day I ran for an hour was pretty amazing and when I started running I couldn't get to the end of the street. (Certainly the many programs online are a great tool)
I would advise you to consider:
watching some videos (plenty of them on youtube) about the proper techniques to run; doing little exercises beside plain running would provide you with more natural, more energy efficient and cleaner movements.
buying yourself proper shoes. Pair of good shoes make huge difference (at least for me), because you get less impact on your feet from surface and proper running shoes should last longer also. Shoes similar to ones like these below would be great; be careful when picking, because lots of 'tennis' or even 'basketball' shoes look similar and those are not optimized for pure running in strait motion (nice scientific answer about shoes here).
eating at least certain time before running or don't eat before running at all. Find out what fits for you; maybe 45-90 mins before if you do.
taking something for drink (preferably not just water). I prefer using those multi-mineral & multi-vitamin(!) water dissolvable pills. Couldn't do without them.
finding yourself proper place to run - preferably, you don't ever want to run on too hard surface and if you could get to run on athletic stadium it would be perfect, because surface there would be good to your legs (try and see!) Check nearby parks or woods (if near) for some flat and softer walking paths; regular earth with low grass for example (other flat and soft surfaces would also be good).