An easy way to test if your chain is too tight is to lift the rear wheel off the ground and give the pedals a kick/spin. If you can see it slowing down at one part of the revolution, or if it doesn't spin for a long time, your chain is way too tight.

There should definitely be a little sag in your chain, how much you can run depends on your bike and riding style. Play feels gross, especially when you're learning to trackstand and such, but it shouldn't prevent you from doing anything.

Pictures can be misleading, but this picture looks like an example of a chain that is too tight:

You can see how the chain appears to curve up from the ring.