How do I use significant figures in my chemistry class?

Significant figures can be annoying, but the fundamental principle underlying significant figures is a very important one: We don't want to lie about the precision of our results.

Significant figure rules are the way they are because we want to accurately represent our results as neither more nor less precise than they really are. Keep this in mind, and the rules will become more intuitive to you.

The simplest way to represent significant digits is to always use scientific notation. Then the number of digits in your mantissa (the part that comes before the 10^x business) is your number of significant digits.

The reason that when we multiply we take the lowest number of significant digits is that this is the overall precision of our result; if we multiply 0.2 g/mol * 0.278 mol, that 0.2 g/mol is saying that we really only know it's about 1/5 of a gram per mole; could be 0.196, could be 0.215; so it wouldn't make sense to keep all three decimals in our final answer. Instead we only keep the 1: 0.06 g, or better yet 6.0*10^-2 g. If we knew it was that precise, we should have said it was 0.200 g/mol, not 0.2 g/mol.

Similarly, when we add or subtract, we only keep decimals up to the least-precise figure, because we don't want to overestimate the precision of our final answer. 1.284 g + 2.9 g = 4.2 g, not 4.184, because that 2.9 wasn't precise enough for us to really know that the result is 4.184.