Autorouters are rubbish. It's definitely worth learning to route and doing the routing yourself. I'm seeing some potential problems with your autorouted board. Tracks coming off pads that leave an acute angle could trap etchant, which will eat away at the copper over time, potentially leading to connection problems down the road. Also, tracks that go off at some odd angle other than 45 degrees just look terrible. I'm also concerned about some of your clearances, though I don't know what voltages are present on this board or what sort of environment it will be used in. Furthermore, I see some illogical routing (unnecessarily long tracks, ugly branches, etc) that could be fixed if you did it yourself.
I cannot express how much I recommend just routing this board by hand. It's small and simple, and the autorouter is just making a mess (and will always just make a mess).