In the first situation, I'm assuming that the coefficient of friction you've quoted is that of static friction, correct? I think the idea here is, how much force can the cyclist apply before skidding? In the second situation, it's obvious the rider is skidding, which means you have to know the coefficient of kinetic friction in order to solve the problem. Or is this problem simplistically equating static with kinetic? In general, static coefficients are greater than their corresponding kinetic coefficients. Are you given both coefficients?

You are given the mass of the bicycle and rider. Therefore, in the first problem, if you were to know the force she applied, you would know the acceleration by Newton's 2nd law! Evidently, this is what you are supposed to find out. Some pieces of the puzzle that I deem important: for regular bicycles, only the rear wheel drives. So, in calculating the amount of force allowed in moving the bicycle forward without skidding, you can use 60% of the normal force in computing the force of friction.

The second problem is a fairly standard problem, complicated only by the 60/40 distribution of the rider's weight. Even then, I don't think the final result will be altered by that fact. You basically only have one force that is active in the direction of motion: friction. So, use Newton's Second Law and integrate.

Does this give you some ideas?