Yep, the couch to 5k or any other run/walk program is a great way to start running. The idea is that you alternate between segments of running and walking, as your fitness increases, the walking segments get shorter, and the running segments get longer. If in the past, some sort of pain or injury stopped your training, it would be worth your while to connect with a running form coach as you start your new routine. Keep in mind, it's easiest to adjust your running form when your mileage is low and your speed is slow.
It sounds like you are trying to do what we call "going from couch to goal", which can be frustrating. To reach your goal of 1.5 miles in 15 minutes, which is 10 minute mile pace or 2.5 minute quarter mile pace, you need to gradually work into it. Your "plan" would include working up to running further than that distance at a slower pace and also shorter distances faster than that pace, so that both goal pace and goal distance are within your comfort zone.
All that said, I would suggest you start off with 20 to 30 minutes sessions where you run some and walk some. At first it may be mostly walking, but as you progress it will become more running and less walking. Run moderately until you feel winded, although not necessarily "tapped out" then walk until you feel you can run again etc. As you build up endurance you should feel better, run farther and run faster. If you are at 12 minute pace, this becomes a 2.5 mile run.
After a week or so, add in some days where you warm up and then run shorter distances at a faster pace...intervals. Maybe the first time you run 4-6 200's (half a lap) and try to do them in about 1:00 to 1:15 pace each with a minute or two rest between. Eventually work that up to 400's (1 lap) in the same pace shortening the rest if you can. You have then run your 1.5 miles (in 6 pieces) at 8-9 minute pace.
The combination of those two workouts over time should make your goal distance and pace very doable, since you have practiced running both farther and faster.
And since you mentioned weight, each pound of fat is about a mile of capillaries in your body. Any weight you lose reduces the structural load on your bones and muscles and reduces the cardiovascular load on your heart while also making your body more efficient at cooling itself.
Lastly, review videos on "proper running form" and technique so that you are in fact running efficiently. Bad form can wreak havoc.
There are a plethora of programs out there to train for your goal, be it Higdon's schedules, the new York Road Runners training schedules or the "couch to 5k". They all are legit and good guidelines for training.
the operative word here is "guidelines". Don't think that you have to follow these training schedules slavishly, without accomodation for the realities of your life. I have found that many people develop injuries when they think they have to follow a set training plan without taking into consideration the reality of their lives. Work gets crazy, kids get sick and sometimes, there are not enough hours in the day to fit in the scheduled 18 mile run. It's Ok. your job is to find find the balance to get the maimum amount done with the least amount of stress. At that point, you will find success and meet your goals.