There were a number of medieval fencing manuals, so we actually have a reasonably good idea of how people fought.

Could for example a sword thrust pierce the chestplate

Thrusting through the chest plate deep enough to injure someone is basically impossible. Cutting through it is similarly impossible.

would you need to look for weaknesses around the harnesses like some movies?

Yes, this is exactly what you need to do. Your targets on an armored man are places like (depending on the details of his helmet and gorget) his eyes and throat, armpits, and the back of the knee. Basically, you want to get your point to where his armor isn't.

How do you do this with a sword? [Well, generally you do it by gripping the blade, with one hand near the middle and the other by either the tip or hilt.](https://en.wikipedia.org/wiki/File:Augsburg_Cod.I.6.4%C2%BA.2_(Codex_Wallerstein)_107v.jpg) This really isn't as dangerous as it looks - swords usually need to slice to cut, or you could percussively cleave through something, but your hand isn't moving with respect to the blade.

Armored combat with swords often turned into a wrestling contest, because it's easier to work your tip or dagger into an opening if the other guy isn't moving much. Hammaborg has a nice series ofn fencing in full plate, and this sparring video isn't bad.

Would a sword be of any use at all, or would you prefer an axe/mace/hammer?

Maces, hammers, spears and poleaxes are all probably a little easier to fight with against an armored opponent.

What kind of a tactic could an unarmored man use to have a chance of defeating someone wearing heavy armor?

Be a much, much better wrestler than your opponent? You have the challenge of having to use armored techniques, while your opponent can use the full breadth of both armored and unarmored techniques.

Additionally, ganging up works well. If you and six of your friends pull someone off their horse, hold him down and work a dagger into his eye, there's not much he can do to escape. Even if he's on foot, if you and a number of friends can knock him down with spears and then pin him, he's also pretty screwed.