G,D&L,
Looking at your profile, I'm guessing that you are a student in a reasonably lightweight type such as a C172 or PA28 - if I'm incorrect in that please tell me and I'll moderate the tone of what I'm saying.
Yes, an ASI (either speedtape or dial gauge) can over or undershoot, but it's small - the devices are well damped and so long as you aren't close to the stall or making sudden airspeed changes will read consistently.
However, the aeroplane itself may well be less well damped and can unless you treat it appropriately tend to hunt airspeed. The reasons for this are two different negative feedback loops - one is the aeroplanes attempts to settle on a specific airspeed / AoA / attitude in the short period, the other is the combination of your brain and arm(s) reading the airspeed indications and trying to achieve a particular airspeed. If you get the two out of sync, you end up with the two negative feedback loops combining into a positive feedback loop and the aeroplane starts oscillating - often quite wildly. This can happen in any axis, but is most common in airspeed or pitch attitude. The technical term for this is PIO or "Pilot Induced Oscillation".
Now the important question, how do you deal with this? Essentially what you need to do is start at a speed above or below the speed you are seeking (if it's on finals, above is best or you're in danger of getting too close to the stall, if you're trying to achieve a particular cruise speed, it doesn't really matter) - get the aeroplane roughly trimmed there. Then very slowly, with no large pitch or throttle inputs (different aeroplanes and pilots favour the two controls for this, but the odds are your light aircraft instructor will favour the yoke), creep the speed down to what you're looking for making very small and relaxed inputs. That way, you shouldn't have a problem with the stick.
If you've got into a PIO, you need to do what a Test Pilot would call "opening the loop", which means stop making active inputs. So, clamp the stick or yoke in the middle and let the aeroplane settle down, or make a single (gentle!!!) input to pitch up or down below or above the required airspeed then start sneaking up on it again.
The other thing is, as JT says, the ASI is only really an approximation to AoA - which is what is really important. So, just as a matter of flying technique in a small aeroplane - learn the attitude for various standard manoeuvres (that is the relationship between the horizon and the aeroplane references - such as the top of the instrument panel), fly to that, then make small changes with the occasional glance at the ASI. Chasing airspeed does you no favours in this context anyway.
G