1 month is not that long. I know it took me a good year until I became a strong cyclist. So, don't be too hard on yourself. Just keep working on your miles and you will get stronger each day.
My breathing is okay but my muscles are tired after 10 miles. Is this normal? Should I be pushing harder? Or just keep adding a mile a week on my daily loop till I get to where I want to go?
Let me ask you, are you muscling through your pedal strokes? If so, you might be in too high of a gear and mashing on the pedals. What you want to do is raise your cadence (a bike computer will give you this reading), up to around 90, and then you will be spinning your pedals.
Also, if you get a bike computer with cadence, you will know when to shift into high/lower gears. For example, if you are riding and your cadence is at 70, then you know that you need to shift down, to make it easier to pedal.
Keep up the great work, and keep plugging away. It takes time and many miles, but you will become stronger each day!
"Life is not a journey to the grave with the intention of arriving safely in a pretty and well preserved body, but rather, to skid in broadside thoroughly used-up, totally worn out, and loudly proclaiming: WOW WHAT A RIDE!!!!"