Yes, that is normal and no need to be worried! Some people's eyes turn to red after strenuous exercises, that is just because of higher blood pressure from the exercises, but naturally, it will turn to normal after 1 or 2 hours later. However, there still have some other reason may cause the issue, some times your eyes will get inflammation with the conjunctivitis, it may caused by the virus that get into your eyes when you cycling. The sweat may get into your eyes with the virus or if you use your dirty hand to rub your eyes. You can relax after running, if after 1-2 hours, the symptom still exist, you may use eye drop for help, if still can not help, please go to the doctor. As for the vision problem, there have some dangerous, you can check if the cornea is broken in the process of sports.
Yes, it is quite normal to have red eyes after cycling and it does not mean you have any vision problem. Cycling, as well as some other strenuous exercises can make the whites of your eyes turn red. Those reds are just bloodshot and they will disappear after you have a good rest. So don't worry about that.