My car is sluggish turning over and won't start battery is fully charged where do I start looking for solutions?

 

Look at the starter.
Sluggish turning over is a sign that either the starter is faulty
or something is binding which is preventing the starter from turning over properly.
or corrosion is preventing the starter from getting the power it needs to turn over the engine

What to do?
Check charge on battery
Check terminals and wires for corrosion.
Corrosion builds resistance and eats up the power needed by the starter
Have assistant turn the key to start and
you take something and tap the starter to see if that helps to crank over the engine
Loosen up the accessory belts and check each pulley for binding

Note: It is also possible that a clogged exhaust will make it difficult to start an engine