Boolean Expression Simplification - Issue I don't understand where to even begin with this problem. Any helpful tips would be nice.
$(A \lor \lnot B \lor \lnot D) \land (\lnot B \lor C \lor D) \land (B \lor \lnot C \lor \lnot D)$
 A: The expression is in Conjunctive Normal Form (CNF), a product of sums.
Probably the easiest way is to create a truth table which lists the expression values for all $2^4=16$ input combinations.  Calculate the truth values for the three parts of the expression. The overall expression can only be true, if all conjunctive sub-expressions are also true.
The analytic way to calculate the minterms of the expression is to resolve/eliminate the parentheses:
Step 1:
$$(A \lnot B \lor A C \lor AD \lor \lnot B \lor \lnot  B C \lor \lnot B D \lor \lnot B \lnot D \lor C \lnot D) \land (B \lor \lnot C \lor \lnot D)$$
Terms with contradictory literals like $B \lnot B$ result in false and are omitted.
Applying the Absorption Law, this can be simplified to:
$$(A C \lor AD \lor \lnot B \lor C \lnot D) \land (B \lor \lnot C \lor \lnot D)$$
Step 2:
$$ABC \lor ABD \lor BC \lnot D \lor A \lnot CD \lor  \lnot B \lnot C \lor AC \lnot D \lor  \lnot B \lnot D \lor C \lnot D$$
Absorbing again:
$$ABC \lor ABD \lor BC \lnot D \lor A \lnot CD \lor  \lnot B \lnot C \lor  \lnot B \lnot D \lor C \lnot D$$
To simplify this sum of products, a Karnaugh map helps for expressions with a handful of input variables:
             cd
       00  01  11  10
      +---+---+---+---+
   00 | 1 | 1 | 0 | 1 |
      +---+---+---+---+
   01 | 0 | 0 | 0 | 1 |
ab    +---+---+---+---+
   11 | 0 | 1 | 1 | 1 |
      +---+---+---+---+
   10 | 1 | 1 | 0 | 1 |
      +---+---+---+---+

A minimized form is
$$C \lnot D \lor \lnot B \lnot C \lor ABD$$

To draw the Karnaugh map right a way without involved reasoning, one can negate the literals of the conjunctive clauses to find out which cells of the Karnaugh map are false/0. The remaining cells are then true/1.
Example: $(A \lor \lnot B \lor \lnot D)$ is false for $\lnot A B D$.
Try this method to convince yourself that it directly arrives at the Karnaugh map depicted above.
