A Good Workout Plan For a Busy Person?
Going for a start on a weekend like you are doing is a perfect idea.
Once you are comfortable with this (sounds like you're getting better already), perhaps the simplest and most time-effective way would be to try and use your bike to get to or from work.

You can start simple: ride to work one day a week, and if that makes you very tired, you can leave it there and return home by other means, riding your bike back the next day. I am a lazy person when it comes to exercise (I would never go to the gym) yet I have found that using my bike to commute effectively gives me 'free' exercise that does not take up any extra time in my day. As a bonus, it gives me something to look forward to in the mornings and evenings too!

No need to rush into things though. As you are doing, start with one day a week, and move from there when you feel ready.