Yes, the symptoms can go away on their own but that doesn't mean that the infection that caused them isn't still there.
As to whether or not they can travel on to your kidneys. . . . .you better believe they do. I became ill with what I thought was a horrible case of the flu; after 3 days of unrelenting pain, fever and nausea I went to the hospital. Where they admitted me for a serious kidney infection. My vital signs were poor; my body was unresponsive to the IV drugs they were pumping me full of and it took me 4 days in the hospital and another 7 travelling back and forth twice a day for IV therapy to clear up the infection. Since then I have had recurrent UTI's that very persistent and hard to cure. It often takes me several rounds of meds before I get feeling better. The thing is I have never had the traditional symptoms of a UTI. There is no burning sensation, no discoloration; etc. My symptoms usually begin with nausea and back pain.
I advise you take your meds; and always be alert to other possible symptoms of trouble. Severe kidney problems can arise from untreated UTI's. I hope this helps.