Step 1: What is hybridization.
Hybridization means mixing of atomic orbitals to make new equal orbitals. These new orbitals are used to form bonds. We must find which type carbon uses in methane.
Step 2: Count the bonds in methane.
In $\text{CH}_4$ the carbon makes four single bonds, one to each hydrogen. So carbon needs four equal orbitals to point in four directions.
Step 3: Count the regions of electrons.
Carbon has four bond pairs and no lone pairs. That is four regions of electrons around carbon.
Step 4: Match regions to hybridization.
Four regions need four hybrid orbitals. To make four orbitals we mix one s orbital with three p orbitals. This mixing is called $\text{sp}^3$.
Step 5: Why not the others.
The $\text{sp}^2$ type makes only three orbitals and sp makes only two. These are too few for four bonds. So they do not fit methane.
Step 6: Final choice.
Four equal bonds in a tetrahedral shape come from $\text{sp}^3$ hybridization.\[ \boxed{\text{sp}^3} \]