Finally got this DS18B20 working. Actually three of them. And in two modes. Parasite power mode and main mode.
A 4.7K ohm resistor is the key.
Starting with the main mode, three wires are needed, as this sensor has three pins.
Datasheet is here http://datasheets.maxim-ic.com/en/ds/DS18B20.pdf.
Connection is almost simple.
Pin 1 to Arduino ground
Pin 2 to Arduino digital input
Pin 3 to Arduino 5V
And a 4.7K resistor between pin 2 and pin 3
Works with three wires, as seen on the followind illustration.
Checked out my local electronics store http://www.oomipood.ee yesterday and got some stuff.
First, a breadboard and some wires. Also got me a soldering iron. Those are just for testing. At least I do not have to twist wires together now.
Then I thought, LED-s are cool for something, right? There is a nice site http://www.ladyada.net/learn/arduino/lesson3.html. So I did get 10 1/4W 1K ohm resistors, and some leds, specifically 2 red, 2 blue and 2 green. 3mm and somewhere near 2000-4000 mcd.