Be warned. Getting fit can be addictive and expensive. I like to run but outdoor running in the Utah winter, well I purchased a treadmill $1200. For Christmas my 14 year old wanted a weight training set, $250. I used it much more than he did and found that I really like weight training. The $250 set became grossly inadequate by my birthday in May. My wife got me a Nordictrack set to the tune of $800. I took to biking to work this past year (12.5 miles each way and a great workout). I have had the bike for a long time but it is in the $400 range. Add a bike computer, light, rack, saddle bag, hydration pack, replacing shifters, gears, etc. and the bike is a few hundred dollars a year. But that's cheep considering what a wonderful thing biking to work can do to your positive mental outlook for the day.
What I've read seems to come down to this. For the average "out of shape" person start with 30 min every other day. In other words start slow. It is OK to feel sore or stiff the next day, but pain means you over did. If you keep with it your body will adjust to the increased demand and the soreness will go away. In other words " No Pain No Gain" is not true. Pain means you damaged your muscles.
There are two main fields of exercise, cardiovascular and strength training. Either should improve your breathing and lung capacity, but I would think cardiovascular would make the biggest improvement in breathing.
I do cardiovascular one day and strength the next, an hour each day. I exercise six days a week (like the song- never on Sunday). I like cardiovascular so much that to slip in a late night 30 mile bike ride on a strength training day doesn't bother me much.
When I started I was bad out of shape. I have not reached my goals but feel great and look so much better.
Let us know what you come up with.