While putting dishes from dish washer into the cabinets my arms got tired! So time to go back to gym and exercise! If I want to start for the first few weeks with moderate running in target heart zone of my age (32-M) for about 1 or 1.5 miles and also do some some sets of weight lifting that I can do 20lb to 25lb, Is that correct to do these two together in one day? or should I increase the running to 3 miles and one day only do that 3 mile running, the other day do just like half a mile running for a warm up and then just do more lifting?
At the level you're talking about, you can do all of it in one day. There's a related answer that discusses how to breakup weight training and running once the two start to interfere with each other. Primarily running and lifting tend to collide with lower body lifts.
Also, you may want to consider doing a real strength training program such as Stronglifts 5x5 or Starting Strength. Either will give you phenomenal results compared to just randomly lifting a dumbbell here and there.
All of your proposed plans sound fine. The only reason to separate your lifting from cardio work would be if you gained access to heavier weights. If you were lifting close to bodyweight then splitting the running and lifting into separate workouts would help you focus on each one better. As is, combine or don't combine them according to your whim.