What exactly is the use of $R_c$, $R_e$ at the emitter and collector? Can we use a transistor without them?

For transistor amplifiers, these resistors affect the amplifier properties, i.e., the gains and impedances (we saw today that for the CE amplifier example, the gains and impedances depend on $R_e$ and $R_c$. We will see other examples.) You choose these resistors to select the amplifier properties you want.

In some cases these resistors can be omitted. You can make a CE amp without an emitter resistor; to find the properties you just set $R_E=0$. (It turns out that it's actually a good idea to put the emitter resistor in there to make the biasing more stable; if it's not there, the base resistor choice becomes highly dependent on $\beta$, which can vary a lot from transistor to transistor. But if you use an emitter resistor you end up with biasing voltages less sensitive to $\beta$.)

You don't really want to omit $R_c$ for a CE amp (and a CB amp, as we'll see) since that would just put the output at $V_{CC}$ which would not be useful. But we'll see later that in the CC amp case there's no collector resistor.