Running is great!

I don't know your level right now but you could try something like this to start running.
Try running for one minute and then walking for a minute. After one week, move to running for two minutes and walking for a minute. Increase the running component by one minute each week after that, until you are up to twenty minutes of running.
Beginner's Running Program
Week 1- Run 2 minutes, walk 1 minute -do this for 21 minutes total.
Week 2- Run 3 minutes, walk 1 minute -do this for 20 minutes total.
Week 3- Run 4 minutes, walk 1 minute -do this for 20 minutes total.
Week 4- Run 5 minutes, walk 1 minute -do this for 24 minutes total.
Week 5- Run 6 minutes, walk 1 minute -do this for 21 minutes total.
Week 6- Run 7 minutes, walk 1 minute -do this for 24 minutes total.
Week 7- Run 8 minutes, walk 1 minute -do this for 20 minutes total.
Week 9- Run 10 minutes, walk 1 minute -do this for 22 minutes total.
Week 10- Run 11 minutes, walk 1 minute -do this for 24 minutes total.
Week 11- Run 12 minutes, walk 1 minute -do this for 26 minutes total.
Week 12- Run 13 minutes, walk 1 minute -do this for 28 minutes total.
Week 13- Run 14 minutes, walk 1 minute -do this for 30 minutes total.
Week 14- Run 15 minutes, walk 1 minute -do this for 16 minutes total.
Week 15- Run 16 minutes.
Week 16- Run 17 minutes.
Week 17- Run 18 minutes.
Week 18- Run 19 minutes.
Week 19- Run 20 minutes.
This of course is a beginner program. You can always modify it if it's too easy. Skip the first two weeks if you can already do that for example. For example from week 15 - 19 you may be able to just push yourself to run the 20 minutes instead of adding one minute a week. It also depends on how often you run. Start out with 3 days a week. Once you can run for 20 minutes you can start adding in
HIIT training. INcreasing the speed slowly like you said is a great idea. Just be careful starting out running is a great form of cardio but it is also easy to injure yourself if you do too much too soon. Your body and joints need to get use to it.
Good luck

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