user38122: It's the first time I use an ESP8266 and I didn't manage to get a proper response in the Arduino serial monitor. I only see weird characters, when I move the 3.3V wire.
You can see the connections in the following picture:

The code I run is the following one:
#include <SoftwareSerial.h>
SoftwareSerial BT1(3, 2); // RX | TX

void setup()
{
    Serial.begin(115200);
    BT1.begin(115200);
}

void loop()
{
    String B= ".";
    if (BT1.available())
    {
        char c = BT1.read();
        Serial.print(c);
    }
    if (Serial.available())
    {
        char c = Serial.read();
        BT1.print(c);
    }
}

Maybe it's something really stupid. I am a total beginner, so I appreciate any help.
EDIT: I think power supply is 3.3V
SOLUTION: the problem was the power supply. I tried exactly the same using the 3.3V from the arduino instead of the power supply and it works perfectly!

user10394: You may well have killed the ESP by supplying it 5v, you have to have a logic level shifter, as Juraj says.
Assuming it's not dead then your next problem is there have been posts about that hardware setup and SoftwareSerial doesn't work at high speed. I can't tell you what it is that doesn't work because turning the speed of the link cures the problem, try 9600 maximum.
[Edited to be less accusational, which was not the original intent]
[Update]
Please see the datasheet here:
https://www.mikrocontroller.net/attachment/231858/0A-ESP8266_Specifications_v4.pdf
The table in section 8.4.4 says that the maximum input voltage is 3.6V
Just after that it says all digital pins are "protected from over-voltage with a snap-back circuit". This snap back voltage is typically 6V.  So does that mean the GPIO are safe up to 6V?
(It might be that this datasheet has been superseded, I am unable to access the EspressIF site at the moment).

38122: <p>It's the first time I use an ESP8266 and I didn't manage to get a proper response in the Arduino serial monitor. I only see weird characters, when I move the 3.3V wire.</p>

<p>You can see the connections in the following picture:</p>

<p><a href="https://i.stack.imgur.com/KwsMQ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KwsMQ.jpg" alt="enter image description here"></a></p>

<p>The code I run is the following one:</p>

<pre><code>#include &lt;SoftwareSerial.h&gt;
SoftwareSerial BT1(3, 2); // RX | TX

void setup()
{
    Serial.begin(115200);
    BT1.begin(115200);
}

void loop()
{
    String B= ".";
    if (BT1.available())
    {
        char c = BT1.read();
        Serial.print(c);
    }
    if (Serial.available())
    {
        char c = Serial.read();
        BT1.print(c);
    }
}
</code></pre>

<p>Maybe it's something really stupid. I am a total beginner, so I appreciate any help.</p>

<p>EDIT: I think power supply is 3.3V<a href="https://i.stack.imgur.com/OBmjx.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OBmjx.jpg" alt="enter image description here"></a></p>

<p>SOLUTION: the problem was the power supply. I tried exactly the same using the 3.3V from the arduino instead of the power supply and it works perfectly!</p>
 10394: <p>You may well have killed the ESP by supplying it 5v, you have to have a logic level shifter, as Juraj says.</p>

<p>Assuming it's not dead then your next problem is there have been posts about that hardware setup and SoftwareSerial doesn't work at high speed. I can't tell you what it is that doesn't work because turning the speed of the link cures the problem, try 9600 maximum.</p>

<p>[Edited to be less accusational, which was not the original intent]</p>

<p><strong>[Update]</strong></p>

<p>Please see the datasheet here:
<a href="https://www.mikrocontroller.net/attachment/231858/0A-ESP8266_Specifications_v4.pdf" rel="nofollow noreferrer">https://www.mikrocontroller.net/attachment/231858/0A-ESP8266_Specifications_v4.pdf</a></p>

<p>The table in section 8.4.4 says that the maximum input voltage is 3.6V
Just after that it says all digital pins are "protected from over-voltage with a snap-back circuit". This snap back voltage is typically 6V.  So does that mean the GPIO are safe up to 6V?</p>

<p>(It might be that this datasheet has been superseded, I am unable to access the EspressIF site at the moment).</p>
