Question: I want to use a formula that always takes the present value listed in a column and then subtract the current row value of column A and add the current value of column B to it. i.e. C4 equals C3 - A4 + B4

How do I copy a formula into Column C so that the present value of column C equals the previous value of colunm C - the present value of column C - present value of column A + present value of column B : i.e. =C3-A4 + B4 with the next row being =C4-A5+B5. Like in balancing a checkbook.

Spreadsheets do not know something as a “present value” or a “previous value”. They are not “history aware”. Any value is a present value in this sense.
If you mean the bottommost value in column C to be the “present value” in the sense of the “last already calculated value of a series to continue bottomwards” and expect the cell one row below to show the next “present value” as soon as the respective rows in columns A and B got a numeric result (or content), you may use the formula:
=IF(AND(ISNUMBER(C3);ISNUMBER(A4);ISNUMBER(B4));C3-A4+B3;"") and fill it down as far as column C shall be prepared.