Taylor, you only need a phono preamp for now. What it does is it takes the signal from your turntable and amplify it while also equalizing it to a line level. Your turntable is not outputting line level output through the RCA jacks.
Your chain should be
Turntable > Phono Preamp > old computer speakers or active speakers or amplifiers
Your chain should be
Turntable > Phono Preamp > old computer speakers or active speakers or amplifiers