I was just wondering because I was thinking about getting a dedicated graphics card and stop using the integrated one since it isn't great
Yes, a 3400g will work with a 1660ti no problems.
What PSU do you have?
600w is fine.
What is the make and model of this PSU?
I am looking at it and it says thermal take smart series 600wWhile Corsair, EVGA, and Thermaltake have many very good power supply models, they also all make some models I would not pair with a 1660ti.
Turn your PC off and take the side panel off. See what model and brand of the PSU are. It ought to say on the side.