When I first started cycle commuting in my mid-20s I had a 5 mile (flat) commute. I was not overweight and no existing health issues but I didn't get any real form of exercise. At the end of my commute I took a sidewalk through an underpass that I was initially afraid of traffic-wise, and as it started coming up a very slight incline, I would often pull over and collapse in the grass red-faced and huffing.

Now I go on 200 km bike rides for fun So yes the more you ride your bike the easier it will get. I noticed significant improvement after about 2 weeks, with more gradual improvement after that. If you have no significant issues I assume you would be able to ride to town in a few weeks, whether you could do it daily or whether it would tire you out a lot or not remain to be seen

A couple of ideas:
- try to avoid potholes Obviously if its a really crappy road they could be everywhere but generally stay about several feet out from the edge of the road to avoid the worst spots
- make sure your bike is properly fitted to you, an improperly fitted bike will tire you out quite quickly! This doesn't necessarily mean a new bike but adjustment of seat height, stem lengths etc.
- if your legs are getting tired pick an easier gear
- if your lungs are getting tired because you are spinning like crazy pick a gear with more tension
- headwinds are tiring! if its really windy out take that into consideration and try again on a calmer day
- cold air can also tire you out if you aren't acclimitized to exercising in it
- go at a pace that doesn't leave you out of breath

Anyways more specifics about your tired - tired from pedalling, tired out lungs, generally achy body (which might be bike fit or just getting used to the bike) might help.