First of all, don't be embarrassed at your current level of fitness. I myself am often tempted to not post certain things here because I am not nearly as strong or as fast as many people here, but the thing is, no one here knows me in real life, so even if they were to laugh at what I can't do, so what? And I don't think anyone really looks down on anyone here. I'm pretty sure everyone who comes here does so either to help others or to gain wisdom and motivation for their own fitness or both. The people who really should be embarrassed are the people who want better bodies but are unwilling to ask someone else how to improve themselves and unwilling to put any effort into it. Good for you for actually making steps toward getting in better shape. If you continue working at this, you'll be able to run several miles at a 10 minute pace and be able to run one mile in well under 10 minutes.
It is true---work on what you want to be good at. If you want to be good at running one mile, run one mile. Now, even if you don't care about how fast you sprint or about being able to run over 5 miles, you certainly could gain a lot from working on sprinting 100 yards as fast as you can or running 2-3 miles. At least it might give you some variety from a normal one mile run. But if the mile is what you want to be best at, keep working on that. And if you have an off-day where it takes you 11 or 12 minutes even when you're giving max effort, don't be discouraged.
Others may disagree, but I don't think you should need to weight train. If you are not very muscular at all, running will build muscle up to your level you need it for the running you do. When I started track my junior high of high school, I was very scrawny and had almost no muscle anywhere, but after a few months, my calves were much, much bigger than they had been. However, after that point, my calves and other muscles didn't get bigger because they were already big enough for the running I did. Unless you want to be good at sprinting or want to run medium distances competively, you probably don't really have a need for weights at this point. Running a mile in 8 minutes is going to be mostly a cardio challenge for you, not a muscular one. Achieving an 8 minute mile shouldn't require weight training. That being said, there are strength exercises that could benefit you. But I'll leave it to others to sell you on those.
To get your mile time down, besides just running miles, are there other forms of cardio you enjoy? Do you have opportunities to play soccer or ultimate frisbee or something like that with friends? Something like that is more fun than just running, and it will help your cardio if you give your max effort when you play. How about swimming? There are a lot of ways to improve your heart and lung performance besides running. Keep running, but if there are other things you enjoy more than running that get your heart rate up, certainly do those things too.
What's your diet like? Are you avoiding junk food and deep-fried food? Are you getting enough protein (meat, dairy, beans, etc.), monounsaturated fats (extra virgin olive oil, nuts, etc.), omega-3 fat (salmon, supplements, etc.), fruits, veggies, whole grains, etc.? If you don't already have a good diet, changing to one will certainly help you meet your goal of a faster mile time.
Good luck!
