What parts do I need to buy in order to build a small industrial-like robot?

That... depends entirely on the physical design of the robot. Something small using hobby servos is easiest, but isn't terribly similar to a real industrial robot. Going further up the scale, something like the TechMover Microbot using stepper motors would be next, but plain steppers are still open-loop, so you'll need to deal with that. The next step up the cost/complexity chain would probably use Dynamixel servos or ClearPath stepper servos.

Whatever you do, you'll have to develop a kinematic model, and deal with flex, backlash, zeroing... the water gets deep fast.

I would suggest starting with something like this SainSmart robot, to get your feet wet and familiar with the issues without breaking your budget.

Quote from Hey Hey

I want to build and program the robot as far as possible in the same way as it is done in industry. The robot should have six servo motors for a free choice of the orientation of the effector. Which parts should I buy for this? Should I buy some small servo motors and 3D-print the arm parts and program the controller with an AVR chip? If so, which servo motors should I buy, how should I wire the servo motors, etc? And I don't want to do it with Arduino at all, because I want to learn how something like this is done in the industry.

As a basic beginner you need to start small. Well I did not and spent 10K on a used six axis robot but looking back I should of started small.

Have a look here,

https://www.anninrobotics.com/downloads

Chris has plans for a open source small six axis robot. It can be 3d printed or you can buy an aluminum part kit.

I've gotten this one for my last birthday!

_{Unfortunatly I haven't had much luck with it yet.}

luck?

Quote from panic mode

luck?

Okay! Okay! I'll admit it! I've been a bit _lazy, just promise you won't tell anyone!

All joking aside, I have assembled it already but I've been focusing on other projects. I'm honestly unsure how I'm going to get all those motors to work yet; it's a bit overwhelming, especially because the OSEPP Arduino Motor Shield 6612 that I got for it and a few other motorized projects doesn't seem to want to work for me with the OSEPP provided class. Rather than have to deal with the headache of figuring out whats wrong with the code right now, I've decided to make my own motor driver (for a different project) just so I don't have to deal with it's nonsense, unfortunately that has not been unproblematic but I see it as a way to simplify everything. Then I'd at least know what is going on with the code because it will be my code.

ALL of that said the instructions are very iffy (because when aren't they?) and you'll have to pay close attention. You might also want to consider the "Board & Controller Bundle" version of the same arm. I don't know what all that bundle comes with, but if you can spare the extra cash it might pay off and get you a working robot arm faster.

I hope this gives you some ideas.

actually i don't see how TB6612 could work with this arm since it is using servos and not steppers or brushed DC motors.

servos require PWM high accuracy pulse control. motor power does not go through external driver chip, only signal which is a weak but accurately timed.

TB6612 does exactly opposite - it is a motor driver meant to drive DC or stepper motor motors. for them high current passing though driver chip.

the two don't go together.... driver is either for weak and fast signal only, or it is high voltage/current device but - significantly slower.

TB6612 is the later one.

to drive hobby servos which are used in this arm, you need different type of drive, either using GPIO directly (no shield) or through shield like https://www.amazon.com/KEYESTUDIO-16-…0NsaWNrPXRydWU=

Quote from panic mode

actually i don't see how TB6612 could work with this arm since it is using servos and not steppers or brushed DC motors.

servos require PWM high accuracy pulse control. motor power does not go through external driver chip, only signal which is a weak but accurately timed.

TB6612 does exactly opposite - it is a motor driver meant to drive DC or stepper motor motors. for them high current passing though driver chip.

the two don't go together.... driver is either for weak and fast signal only, or it is high voltage/current device but - significantly slower.

TB6612 is the later one.

to drive hobby servos which are used in this arm, you need different type of drive, either using GPIO directly (no shield) or through shield like https://www.amazon.com/KEYESTUDIO-16-…0NsaWNrPXRydWU=

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My other project is dealing with some DC motors. I just haven't touched the arm project because I am wrapped up with the DC motor treads project.