You have an upright riding position. Cute bike, btw. I was looking at the Dawes bikes on Ebay awhile ago but decided to get a used road bike instead. I'm going to echo others and say ride and increase the distance over time.
It didn't take me too long to be able to go more than a few miles when I started riding again. I hadn't ridden a bike in well over 10 years, and I was far from regularly active, or healthy for that matter. I started on a mountain bike and within a couple-few weeks I was able to do 12-15 miles comfortably. I rode 3 to 4 times a week, give or take. On the mountain bike I was only going between 5-12mph though, depending on the section I was riding (hilly area here). The speed & distance increased quite a bit and was much easier once I switched to a road bike.