You have to be really careful in the heat. I just had a conversation about this with my husband yesterday. (We are in GA and it has been hitting 100 all this week, with high humidity, too). I just started running this past January and I was really starting to get into it with doing some speed work (to try to cut down time on my 3 miles that I like to run, b/c it just takes too much darn time to get the run in at my current speed, not because I want to do races necessarily). We discussed checking into a non-contract gym with a treadmill, since I don't have anywhere to put one right now. The bedroom has the total gym in it already, the garage has the Powertech system (not that I want to do treadmill work in the garage anyway---that's no improvement over outside).
If you have the space, I would go for a treadmill, or perhaps an elliptical.
As for actually getting up to "running" vs. walking... I didn't do Couch to 5K, but I have heard it recommended highly by several people. I just did my own... sort of a fartlek training program, lol.
Jog really slow until I couldn't, then walk. As soon as I wasn't breathing too heavily, jog some more. I began counting each step/stride on one foot. Every time my left foot hit, for example, I'd count, and then I'd set goals for myself. If I could only run for 60 strides, then I'd push to 65. When I could do 65, I'd push to 70. Eventually (in a very short time) I was counting "minutes" that way. Running for 2 minutes (120 strides--I know, not exact), walking until I could breath. Running for 90 seconds, walking. Etc. Then in what seemed like nothing, I was running for 700 seconds/strides. Then it became mind numbing to count, and I started using my wristwatch. I used my car's odometer to measure the distance to certain landmarks and use those as goals. When I finally could run .7 miles without stopping I was THRILLED. Then I pushed really hard one day (too hard) and went 1.7 miles without stopping, and felt like I was going to throw up. My husband said to slow down, I was running too fast. (not really.. but for ME, it was) I did slow down (wasn't sure it was possible to go any slower than I was!), and I went an entire 2 miles. Since then, I don't push too hard. I did fartlek sprints and then slow jogs.
But, in the heat the last week or so, I haven't been able to get out. I am NOT a morning person, and by the time it is "cool" enough (below about 88 for me) it is too late, because it is almost dark outside.
You CAN teach/train your body to "run" it all without walking, but walking is part of the training... just keep going when you have to walk. Don't stop moving or your legs become dead weight.
And with the heat where you are, I would def. go with treadmill inside... not waiting until the evening or messing with a stroller. I know some people do the strollers, but for me, that affects your stride and exertion level, and in the heat where you are, and a relative "beginner" (not unlike myself) I think you would be pushing yourself to do too much too soon.
Just having my well trained dog on a leash affects my stride, because I am gripping the leash. :( So it is def. something to consider...