I find it difficult to understand as to why current behaves differently in an RL and an RC circuit. At time $t=0$, in an RC circuit the current is maximum, whereas in an RL circuit its $0$. Why is it so? The current should rise from $0$ to some value in both the circuits, so where am I going wrong?
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Circuits with inductors are sensitive to changes in the signal - think of them as differentiators. Circuits with capacitors are responding to the integral of the signal over time.

When you first turn on a circuit, the current wants to make a step change - which the capacitor doesn't care about, but the inductor resists vigorously. Thus the current will flow through the capacitor without any problem (there is no built up charge so no potential resisting the current flow), but the inductor will not permit the instantaneous change in current (back emf given by $-L\frac{dI}{dt}$).

So current in the capacitor starts big, then drops off; in the inductor, it starts small, then builds up (at some point the current in the inductor is only limited by the series resistance; while current through the capacitor goes to zero once enough charge is built up to get a potential equal to the applied potential).
1. Charge on the capacitor does not change instantaneously or suddenly if there if a resistance in the path(series) of the capacitor.
2. When an uncharged capacitor is connected with battery then its charge is zero initally hence potential difference across it is zero. At this time the capacitor can be treated as a conducting wire.
3. The current will become zero finally(that means in steady state)in the branch which contains capacitor.