So I guess I just buy the components and solder them in? Correct?
That's pretty much it. Watch the polarity of electrolytic caps. And the orientation the IC's. You may want to order a couple IC scokets in case you may want to 'roll' the chips later.
A cheapo Radio Crap iron should be adequate for your first project. But if you are sure you'll being doing more projects down the road, then invest in a better iron. If tangent recommends the euthaic, then it must be good.