The heat equation is generally solved by separation of variables: Assume a solution of the form: . That is, a product of a function of x and a function of t. Now, substitute the expression into the PDE and obtain separate ODEs in terms of t and x. Solve those ODEs under suitable "well-posed" conditions (boundary and initial conditions) to arrive at a solution. I'd recommend "Basic Partial Differential Equations" by D. Bleecker and G. CSordas. Here's a well-posed problem:
If you get into PDEs, always try to "pose the problem well" (correctly define the boundary and initial conditions). The style of these will change with the PDE.