Q:

How to play a Tank Destroyer in World of Tanks?

I've played several Tank Destroyers, SPGs and Light Tanks so far and reached Tier V with each of them. (Panzer IV, M41, StuG III)
My observation is that the overall difficulty level increases because you are faced with tanks which are more tough nuts to crack. In case of an SPG I had to learn how to properly aim at the weak spots and/or calculate with tank speed (I noticed that the higher the tier of the tank the opponent sits in the more aware they get of a possible SPG threat). I also get bigger guns (155mm howitzer in M41 for example)
In my Light/Medium tanks I had to learn how to use cover so I don't die instantly and how I can use my speed to work myself around bigger threats. I do feel that the higher tier I get my tanks become more capable.
In case of my tank destroyer StuG III I feel that this is not the case. I'm not as fast as a lighter tank, and I do not have the firepower of a heavy tank. (I can't shoot behind covers either). The StuG III has the same derp gun as the Hetzer but since it is a higher tier I often face Tigers and KV-1s. I also noticed that since the StuG III's armor is not as sloped as the Hetzer's I am more vulnerable. I don't feel the same progression with tiers as I do with other tank types. It seems to me that I did not learn something important about playing tank destroyers since the higher tier I get the less fun I have with them.
Do you have some guidelines for a puzzled tank destroyer player?
StuG III specific answers are also welcome

A:

Many tank destroyers, and particularly the Stug III have a great camo rating, which means that if you play carefully the enemy team will have great difficulty spotting you. 
While playing this particular tank, and other TDs like it, try to stay behind the rest of your team and lay down supporting fire form a sniping position. 
If you're not familiar with how camo and spotting works, I recommend this camouflage tutorial. 
Some general tips:

Stay behind a bush, but remember that firing your gun will temporarily remove the additional camo value of the bush
Equipping your TD with a camo net, and training the crew in camouflage will greatly increase your overall camo rating. 
Remember that moving your tank, even by slightly turning it lowers your camo value. With TDs and SPGs you can prevent accidentally turning your tank by locking it in place by using the (default) X key. 
The commander's sixth sense skill is great on TDs as it will help you know when you're spotted.
The Stug III is responsible fast and mobile, if you are spotted or think you might be soon, relocate to a new position from which you can safely snipe. 

A:

As an alternate answer to the excellent camo-oriented answer @Xenox provided, there are also tank destroyers that do not rely on camo:

British TD starting from the AT range
German TD starting from Jagdpanther 2/Ferdinand
US TD starting from T28/T28 Prot

Generally there are 3 types of tank destroyers:

Camo-oriented: They are usually low profile, and turret-less
Maneuverable: Either with turret, or very good turning speed
Heavy armoured: slow, above average frontal armour

The camo one, @Xenox has answered.
For maneauvrable TDs (e.g. E-25, German turreted TD line...etc.) It's all about positioning, dishing out as much damage as you can, then withdraw and re-position, rinse and repeat. So learning how, and when to flank, learning paths to take so you can quickly retreat and move somewhere else, is the most important aspect for these kind of TDs.
Heavy armoured TDs (JP-E100, T110E4, T95...etc) is about finding the spots that conceal your weakpoints, avoiding being flanked, whether by positioning, or knowing where your team mates are and how they can protect your flanks. Knowing when to push forward is also very important due to the slow movement speeds of these tanks, not being at the scene of an important skirmish can determine a match's outcome. As a heavily armoured TD you can also take the role of drawing attention, since your shots hurt alot people tend to want to take down TDs as quick as possible, so if you can utilise your armour well, you can, for example, draw attention away from your flanking team mates.
There are numerous ways to play a TD, so it's not restricted to what I said above, e.g. a heavy TD can also play camo-oriented, if they can find bushes that are big enough.

