Could you ride a standard bicycle on the surface of mars?
Yes, you could. You would balance a little differently, because the force of gravity is a little different, but the force is similar because it is still in the same direction (down, i.e., towards the center of the planet).
The main difference is that the acceleration due to gravity on Mars is about a third of what it is on Earth, so you would feel lighter, as would the bike. Also, the bike would tip slower.
You can understand this latter fact by considering a pendulum. (Balancing on a bike is similar to balancing an inverted pendulum.) For example, on earth, a pendulum of length 1 meter will swing back and forth with a period of approximately (in unit of seconds):
$$
2\pi \sqrt{1/9.8} \approx 2
$$
Or mars the pendulum period would be approximately:
$$
2\pi \sqrt{1/3.7}\approx 3.3 \;,
$$
which is larger than on earth.
There would be other difference as well, for example you could spin your wheels easier on Mars, but the differences would not be insurmountable.
There are low gravity physics driving simulators that you can play with to see some of the differences. Most are built for cars, but bikes are similar except that you have to balance a bike in the transverse plane (as discussed above).