An audio chain usually consists of a source (turntable, digital DAC, FM tuner, tape), preamplifier (sometimes optional), amplifier, speakers. An “integrated” amp has both a preamp+amp, and sometimes includes a DAC, streamer, phono stage (phono preamp). If a digital source, you’ll need a streamer, a DAC, and a digital file source (CD/SACD disc, Tidal, Qobuz). Analog you’ll need a turntable, tonearm, cartridge, and a phono preamp.
Cabling we’d usually suggest at least replacing stock power cords with quality cabling at least to the level from Blue Jeans or Audio Envy, but may be outside of your budget.
Speaker efficiency determines the amount of amplification required by the amplifier. Each 3db of change means a doubling of sound pressure. You can learn more about it here. Average speakers are somewhat 87-low90s db, and efficient ones above that. Unfortunately your speakers are less efficient and thus requires more power to drive which may include ample current needs also.
$500 is a very small amount, not much available at this price point, doubtful if you can get beyond box store sonics. TAS (The Absolute Sound) annual list starts at $800 and goes to $150,000. I doubt at this budget you’d get anything high end sounding. Quality internal parts cost money, especially quality transformers. Maybe if you can get your budget up something like a used Hegel integrated so you can hear what your speakers can do with quality amplification.
+1 the book “The Complete Guide to High/End Audio” by Robert Harley (editor in chief for The Absolute Sound magazine. It’s like an illustrated detailed encyclopedia - no other publication comes close