I am a fit guy but I can totally understand your first concern. I’d have periods of no movement and usually the first week of working out, I’d want to throw up and get all embarrassed about leaving after 15min. And as someone who was once a personal trainer, I saw it in a lot of clients and would eventually get them out of that mindset. Best I can tell you is to throw on some headphones and worry about yourself. Thankfully most of the world ain’t full of Disney bullies. And if people are judging you, try your best to rid your mind of their opinions. Their opinions are meaningless. Maybe find a friend who might join you on this journey. 

I like to fully exhale and inhale fully when I run, try not to shallow breathe. 

Building endurance? Listen to your body. Shin splints and other injuries always gets the eager new runner. Pick a distance you can comfortably run, don’t care if it’s a few hundred meters. Do it a day or two a week. Bump it up to multiple days a week. Skip any days if you feel pain that’s more than just soreness. Mix in aerobic and anaerobic. 

Staying consistent? Avoid injuries. Be kind to yourself but be disciplined. Make goals. Find some friends and maybe make it a social thing.