For addition and subtraction, you have to break one of the rules that you have undoubtedly always been taught: carry all of your decimal places, and only round in your final answer. When using significant figures in these calculations, your final answer can only be as precise as your 'least precise' value.
Here's an example:
1.23 + 4.567 + 89.1011 + 121.3
In this example, the value with the least precision is the final one: 121.3. It only has one decimal place. As a result, your final answer CANNOT have any more precision than one decimal place. You can't be sure if it is 121.300000, or what is beyond the 3, and so your degree of precision must be just the one decimal place. Therefore, you perform this calculation as it appears, but you must round your final answer to one decimal place. In this case:
1.23 + 4.567 + 89.1011 + 121.3 = 216.1981 = 216.2
Subtraction calculations would work the same way.
For these types of calculations (addition and subtraction), it must be pointed out that the number of significant figures in your final answer does NOT depend on the number of sig figs in the original values.
As I mentioned above, you don't always have to carry your decimal places to the end. If this were a 2 part problem, perhaps like this:
1.23 + 4.567 + 89.1011 + 121.3 = x
x + 3 = y,
you would solve the first line to be x = 216.2, and then substitute this value into the second line, and add 216.2 + 3, which would give you the final answer of 219 (applying the same rule as above). You wouldn't have to carry all your decimal places over to the second part.
Hopefully that is clear enough of an explanation for addition and subtration.
For multiplication and division, the rule is the opposite. That is, the number of significant figures in your final answer is equal to the smallest number of significant figures involved in the calculation.
Here is an example:
5.56 x 2.998 = ?
We can see, based on the rules of counting significant figures, that there are 3 and 4 sig figs, respectively, in these two values. Therefore, our final answer must have 3 sig figs. So, instead of 16.66888, we would report 16.7 (3 sig figs). Otherwise, we would imply a degree of precision that was nowhere near what we were working with in the beginning.
I hope these explanations and examples have been helpful to explain some basic math functions involving significant figures. As always, please comment with any questions you might have! :)