It really isn't a question about which bases and nucleophiles do I have to memorize are strong. It's really just a matter of knowing what a base or nucleophile is.Well, what is a base? A base is a molecule that can donate electrons. So, if a molecule has a negative charge on it, it is probably a base. It doesn't have to have a charge in order to be a base, as it could have a lone pair of electrons that do not give it a charge, however in almost every single case this will be a weak base (e.g. ammonia). To determine whether or not it is a strong base, look at the structure of the molecule. Does it look to be moderately stable with the negative charge? Does it have electron withdrawing groups close to the negative charge? Is it resonance stabilized? If so, it might not be the strongest base. If not, then there is a high chance that it is a strong base. Electron donating groups can also indicate a strong base. Sodium ethoxide, the example you gave, is a strong base because there is a negative charge on the oxygen, and the only thing the oxygen is attached to is an ethyl group that is slightly electron donating. So, the oxygen readily donates it's electrons.
A strong nucleophile is basically a strong base that isn't sterically hindered. This is a simplified definition, but it will almost always work.
I hope this helps, and I hope it was clear. I'm in a rush so I had to be quick. Ask if you have any questions.