Like you I didn't run races for many years until I reached my later 30's. However, unlike you I had a longer history and better base (10 miles x 3 days per week for 20 years).
Since I had 3 kids and a 45 to 60 hr/wk job, I had to make the most bang for the buck out of the limited time I had to train. This is what I did at each level and the impact on my times:
1. Base mileage year round: 3 run 3 days per week, 10 miles per run. Before I started racing, that is. My tempo pace was around 8:45 to 9:00, if that.
2. In the fall racing season and also spring racing season, I would start doing longer runs (up to 13 or 15) on the weekend. Also, I would do a tempo run once a week over the full distance, and after a few weeks of training 2 out of 3 runs were tempo runs. Plus, I would do one race of 10 miles or longer at least every 3 weeks. The best race pace over half marathon was only 8:00 per mile doing only that.
3. After plateauing on the above program, I started doing 1/2 mile repeats once a week during racing season. On a typical run, warm up distance, then 1/2 mile as fast as I could, but still able to jog for 1 to 2 minutes afterwards, then do another one, then another, up to 5 repeats. Then after 2 weeks I could do a half dozen one mile repeats, with similar recovery jogs in-between. After a year, this program shaved my half marathon pace down to 7:30 pace roughtly.
4. Scratched 10 to 12 miles x 3 days per week (30 to 36mpw), instead upped it year-round so I ran 12 to 15 miles every time I went out, off-season (38 to 42 mpw, generally). Usually 2 twelve milers and a 14 or 15 miler on the weekend. During racing season, I lengthened the weekend Sunday run to 16, 18, 20, 25, depending on race upcoming, and did a shorter run on Friday before the long run which had the 1 mile repeats (up to 9 consecutive repeats at 5K pace). In addition, I did 1/2 mile uphill interval repeats (3 or 4) if the regular intervals needed to be further challenged.
5. Did upper body weights at home 3 session days per week, 15 reps to failure times 3 sets per session. Also, swam 100 laps one day per week.
6. Combining (4.) and (5.), I got my half-marathon pace down to 7:00 min pace and slightly below.
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In conclusion ---
The effect of base mileage cannot be underestimated, as previously stated you need to get this down first.
Second, go long if you don't have the time to go every day.
Third, year round pick one day per week for a tempo run -- fast over the whole long run, but not dying. Just make sure you push it a little the whole way.
Fourth, during your racing season, add in one day per week of 1/2 mile to 1 mile repeat intervals, hard enough so you can barely jog afterwards, but easy enough so you can punch out another one after a 2 minute recovery jog. Put everything you have into them. Work up to being able to do nine 1-mile repeats --- this may not be possible your first year. Start the intervals about 5 weeks before your first race of the season, Then about 10 days before the race, reduce the number of repeats and the intensity. For example, mix into your normal run 0.1 mile or 0.15 mile x 4 or 5 repeats once a week in the last 10 days before the race, nothing harder. Other than that, drop your total mileage the week before the race about 25 percent, and don't do a tempo run less than 5 days before a race.