Running- increase distance or speed first?

Congrats on the first mile!

Distance, distance, distance. Work on endurance first. Run at a speed that would allow you to carry on a conversation if necessary.

Speed will come naturally as you progress, and eventually you can add in speedwork drills to help too.

I was running 12 minute miles when I started training for a 10-mile race. I completed that race less than two months later at a 10:18 pace. Now I regularly run sub 10 miles and would love to shave another minute off. I do speedwork, but my longer runs are all about endurance.

+1 on distance /endurance.

Over time your speed will improve by itself as a result of your improved fitness. Once you've found yourself able to run your goal distance comfortably then you may want to build some speed work into your program but for now don't worry about it.

You've probably heard this before but......take your time increasing your distances, most running injuries are a result of trying to do too much, too soon too fast. If you're not already doing it consider adding some strength training to your routine (people are often surprised that upper body strength affects your ability to maintain good form), cross train (bike, swim, row - whatever you may enjoy doing) and spend some time every week working on core, hips & glutes (weak hips & glutes frequently manifest themselves as knee pain).

Most importantly......have fun!

Definitely distance/duration. Gotta have a good base before you start working on speed.

Distance.

I've found that my speed will naturally increase as my base/experience increases anyway.

You can do both. You should increase your distance no more than 10% of your current mileage per week. Its slow but its progress and will limit injuries. Besides running a 1 mile plus a quarter is a lot less daunting than running 2 miles. As you do this, alternate between static runs (remaining at 4.1) and Tempoe runs ( 1 min @ 4.1, 1 min @ 4.3, 1 min @4.1) By doing 1-2 runs a week in which you are alternating the speeds, will help you increase your overall speed. Additionally, get off the treadmill. Get yourself a running watch and get outside. 5k races are held outdoors, it also stops you from getting bored so you can go a lot further wthout even noticing it. Besides that, a treadmill assists you in running, when you run outside, its entirely on you. Good Luck.