Or do I need to do something else? I have one advert on my site (it appears on every page), and I'm wondering whether I just have to leave it there to make me some dollar.
You can have upto 3 ads per page. this could multiple your revenue but potentially affect user experience. And yes thats all you have to do once you have adsense, just put the ad coding in the right spot and check it's showing ads and your all set.
I agree with other replies. We started our site with a certain placement of ads. It was not optimal and we did not know it. After quite a long time, we tried to do A/B testing of various ad positions and ad sizes. The result was incredible and our income from the ads increased dramatically. So, to answer your question, yes, you can just put it there and do nothing and if you have some traffic, you will earn some money from it. But if you experiment a little bit (with position, number of ads, ad formats, ad colors), you might be surprised to earn several times more. However, if you have only very little traffic, experiments are hard, sync the results would not be statistically significant to prove anything ... so if we are talking only about tens visitors per day, it might not be worth focusing on ad optimization right now and rather focus on getting more traffic.
I've been using adsense for 8 years now but I also have never understood whether I should just leave the ad there or try and figure out other stuff such as custom channels. I set up custom channels (with targetable ads) a few years ago but haven't seen any change, if anything I've made less over the last few years. Do they offer any real benefit?