Most of my answer is already contained in this answer, although the questions are not really duplicates.
A couple things that I will reiterate:
- The biggest mistake that the vast majority of runners make is going too hard on their easy days, and not hard enough on their hard days.
- Speedwork is the icing on the cake, make sure you bake the cake first (i.e., get your mileage in before worrying about speedwork).
- Not all speedwork is done on the track.
Make sure you have a base first. For new and returning runners, the best gains are made from the day in, day out putting in the miles. For a non competitive runner, this is going to be the biggest thing. Get out there, get the miles in, recover/rest as necessary. There are many many runners that run every day and do so quite safely.
Once you have a base and are running consistently in the 30-40 miles per week (mpw) range, you can start throwing in things like hill repeats, pickups/strides within your runs, all of which will start helping your speed. At this point you can start working on pace/tempo runs, individualized track sessions (such as 1-2 mile warmup, 8x800 on 2 mins rest, 1-2 mile warmdown), other traditional speed workouts. You'll come across two terms, interval and threshold. Interval work increases your top end speed, threshold work increases the time you can spend at or near that top end speed. Even when you start incorporating speed workouts, the majority of your time should still be spent in steady state running, with the addition of the pickups and strides.
And, if you are carrying extra pounds, the apocryphal lore (Borne out in my own personal experience) is that you will gain 3-5 seconds per mile for every pound of weight you lose. So if you weigh 180 and run at an 8 minute/mile pace, lose 20 lbs and the same effort should get you to a 7 minute/mile pace.