It's best to draw out what happens if you're confused.
Draw the H2O molecule with its lone pairs as well as the H-Cl. How does the H2O become H3O+? When it donates an electron to the H in H-Cl, while the H in H-Cl releases its electron from the bond to Cl, which becomes Cl- by accepting that electron.
So H2O is an electron donor and therefore a base, while HCl is an electron acceptor and therefore an acid.
All Bronsted acids are Lewis acids, and all Bronsted bases are Lewis bases. But not all Lewis acids are Bronsted acids and not all Lewis bases are Bronsted bases.