Galvani set up the first electrochemical cell, Volta demonstrated that it did not require the organic matter of a frog's muscle, but rather could be set up with just brine (saturated sodium chloride solution) soaked cardboard. Between them, they have effectively set up the first electrochemical cell. Volta also demonstrated the value of placing them in series.
Neither of them realised that there was an electron transfer reaction occurring, although they have laid the basis for modern batteries and electrochemical devices.
Davy refined the cell, recognised that the electricity was due to a chemical reaction, and demonstrated that electrolysis - the use of electricity to induce a chemical reaction - was possible.
Davy's contribution is significant - once it is recognised that a reaction is occurring, you can proceed to investigate its nature. Davy was unaware of electrons, as they were yet to be discovered, so he did not describe electron transfer reactions in the sense that we would today. Nonetheless, his work in the area added greatly to our knowledge about them. He:
* recognised that a chemical reaction was responsible for the production of the electricity
* developed improved versions of the Voltaic pile (ie better batteries)
* demonstrated that electrolysis was possible
* prepared samples of the elements sodium and potassium
Faraday continued Davy's work, quantifying it into his laws of electrolysis. He demonstrated that the mass of substance produced by electrolysis was proportional to the quantity of electricity passing through the cell. This he quantified using an instrument that he invented, called the Coulometer. He determined that each element had an electrochemical equivalent, B, that was the constant of proportionality between mass and quantity of electricity. He found that for series with the same valency, like Cl, Br and I, the electrochemical equivalents was in the same ratio as the atomic weight - this was added evidence for Dalton's atomic theory.
Faraday's results are easily understood in modern terms. His first law of electrolysis allows us to predict the amount of products from an alectrolysis experiment from the equation for the reaction, the current and the time. This is the basis fo quantitatively describing electrolysis.
Remember when examining historical developments like this that you cannot expect to look at discoveries solely in the light of modern understanding. The fact that these scientists did not describe electron transfer as we knoiw it does not mean that they did not lay the basis for that understanding. Look at the time in which they worked:
Galvani (b. 1737, d. 1798)
Volta (b. 1745, d. 1827)
Davy (b. 1778, d. 1829)
Faraday (b. 1791, d. 1867)
Thompson's discovery of the electron did not occur until 1897, so none of them knew about electrons. Dalton's atomic theory was not proposed until the early 1800s, and was not widely accepted until some years later. Instruments that we take for granted - ammeters, voltmeters, and high quality equipment for easily measuring small masses were all yet to be invented. Furthermore, Arrhenius' theory of solutions (for which he was awarded the 1903 Nobel Prize for Chemistry), recognising the existence of ions in solution, was not proposed until long after Faraday's death. In fact, Faraday's work laid the basis Arrhenius theory. (The timing of the theory of solutions should also be remembered when discussing theories of acids and bases. There is a tendency to minimise the contribution of Davy, with his theory about replaceable hydrogen. However, if you think about it, replaceable hydrogen is a pretty good description of an acid if you're working at least half a century before the existence of ions is recognised.)
Hope this better answers your question.