I'm confused as to how this works. A stronger acid means that I will have a weak conjugate base, so that would mean that my phenol is a weak acid (pka= 10) and therefore gets a strong conjugate base? How do I know the strength of NaHCO3?
While one could have tables of base strengths, you will not find that to be the convention. However, it is simple to realize that they are related properties. The stronger the acid, the weaker the base. Therefore, if you look up the acidity in a pKa table, you can use that information to predict the relative basicity of the conjugate bases.
In this instance, just as stated, the pKa of phenol indicates it is a weaker acid than carbonic acid, therefore phenoxide will be a stronger base than bicarbonate (the stronger the acid, the weaker the base).
Just one additional consideration. One must be careful with carbonic and similar acids in that it has two values, carbonic acid and bicarbonate. Don't use the bicarbonate value for carbonic acid (this was not mistaken in this example).