I've recently bought a reflecting telescope with a focal length of 650mm, which lens should I use if I want to see saturn and its rings?

Saw it was a 5" (130mm), they all seem to be 650mm FL. Try your 6mm. That will show more detail. Also make sure you are pointing at a planet. Mars starts out low in the sky and will look like a crappy blurry dot till midnight (its just too low and there is too much heat moving around the sky early on). Saturn should look great. Jupiter starts high up, did you try that? You could get a barlow, that would help.

Like many beginners in astronomy, you have false expectations engendered by all those pretty pictures you see online. Basic fact: All the planets are very very small in angular size as seen from Earth. Normally you need a magnification of at least 200x and very steady air overhead to see detail in the planets. With a 130mm aperture, your maximum useful magnification is around 130x; anything more will enlarge the blur, not the planet. With your telescope, the shortest focal length eyepiece you can use will be about 5mm. This will give you images twice as big as your 10mm, but they will still be small. It takes eye training to see detail at that low a magnification, and the best way to get that training is to try making pencil sketches of what you see. That's why we usually recommend 150 or 200mm as the minimum aperture a beginner should consider.

[Edit] I had a similar telescope for a while, and I found that I could use a 4mm eyepiece (162x) on nights when the air was really steady, but this was a high quality eyepiece, a Tele Vue Radian, which costs about as much as your entire telescope! Experienced observers can see an amazing amount with smaller apertures, but that is because of years of eye training. Beginners like yourself need all the help they can get, and mostly that help comes from aperture.