I have found various sources saying that ketones are acidic (pKa=20), and then relate it to the formation of the enolate ion. However, isn't an enolate ion of a ketone basic due to the oxygen's negative charge and all the pairs of electrons surrounding it?
I think you need to recall the definition of an acid (in the Bronsted-Lowry scheme). An acid is a proton donor. A base is a proton acceptor. When a acid dissociates, it forms a hydrogen ion and the conjugate base of the acid. Remember, acid and base are only relative terms. The species which loses a proton in the reaction is acting as an acid and that which gains a proton is a base.
In the example above, the ketone is acting as an acid because it donates a proton. The hydride anion is acting a base because it accepts a proton. The resulting enolate anion is stabilised by delocalisation of the negative charge onto the oxygen. In the reverse reaction, the enolate would act as a base, accepting a proton from hydrogen (this is unlikely to happen though because the hydrogen will be liberated as a gas).
It will shift the places of the double bond and one of the hydrogen atoms, which will end up in a compound with a double bond between two of the carbon atoms.
That makes an enol which isn't as stable as a Ketone. The enol and Ketone are in a equilibrium which causes the enol to lose an hydrogen ion which then - the enol will become an enolate.