What are the V-Speeds used during a normal takeoff in a multi-engine airplane?
$V_1$ is takeoff decision speed.  It is based upon:


Runway length
Temperature
Reduced thrust takeoff
Takeoff weight
Takeoff flap setting


If $V_1$ is higher than $V_R$, then (as far as I am aware), $V_1$ will be adjusted to be the same as $V_R$.  You have to make the decision to takeoff by the time you rotate, so it does not make much sense to have $V_1$ higher than $V_R$.

$V_2$ is takeoff safety speed.  During a one engine inop takeoff we would accelerate to and climb at this airspeed to our acceleration height (a height we could level off at and be safe from obstacles while we further accelerate).  We also referenced certain flap transitions on takeoff to $V_2$ (e.g. $V_2+15$ - flaps 0)

To address the rest of the question, $V_1$ and $V_2$ are primarily used in large aircraft.  In light twins you also need to be aware of $V_{MC}$.  $V_{MC}$ is the minimum controllable airspeed with an engine inop.  In particular you do not want to be slower than this airspeed during climbout in case of engine failure.