I am fairly new to running, but have been training for a marathon since January with a training class (it's NEXT WEEK!!
) so I have a couple tips I learned from class.
First, your pace is going to depend on how far you want to go. For me, since my distance is 26.2 miles, my pace is a lot slower than what I 'could' run. So think of that as you add miles.
The class I am taking suggests that I start out slower and speed up later in the race if I have extra energy. I think this is just because of the distance once again though. They don't want me to crash and burn. For a distance of 3 miles, I would suggest a good 5 minute walk, 5-10 minute slow jog (or finish out the first mile) and then kick it up to the pace you want to be at. If it was a 5K race, I would suggest that warmup before the race and the entire race at pace. Your logic of increasing the speed of each mile as you train sounds good to me, but you do want to get to an even pace overall for a short run as I think it is easier. Once you get in the groove of a pace for that day it is easier to maintain than keep changing pace during a single run....IMO.
As for increasing your speed so you are at the pace you want to be at, the best way is to include speed workout drills. There are specific running activities and sequences that you can do to make yourself faster. These range between trainers, but mainly...mile repeats, hills, cadence drills, etc. I would suggest doing these 1-2 times a week instead of your distance run. This will help you increase your pace over time and make your feet faster.
Finally, there are 'tests' you can take to determine what your pace should be for certain distances. In the class I took, we would run a single mile as fast as we could (without puking or dying) and then enter that time into a math equation. The equations would tell us what our expected race pace is based on that mile. For example.... If you can run a single mile at 8:30 then your overall pace for a 5K would be 9:03, for a 10K 9:47, a marathon 11:03. For you to run 3 miles at an 8:30 pace overall, your single mile time should be about 8:00. If you cannot run a mile at 8:00 right now than you are projecting an overall pace that is too fast for your training. This doesn't mean that you can't get there....just that you aren't there today.
Here is the link I use to determine my pace.
Jeff Galloway's Race Prediction Formulas for the 5K, 10K, Half Marathon and Marathon distances
I hope this helps! Let me know if you have any other questions and I am sure you will get other opinions on this as well. Good luck with your running, but be careful....its addicting!!