Nice amp and speakers! They are capable of better sound than you'll ever get from those vinyl rigs. Improving your vinyl front end is definitely a good move.
Having limited funds, you'll get more bang for your buck from a used rig than a new one. The challenges are to find a rig in good condition and a seller who understands how to pack it properly to avoid damage during shipping.
Finding a local seller would address both issues, as you could demo the rig before buying and transport it yourself. In addition to ebay, other possible sources are craigslist and - of course - Audiogon! The typical Audiogon seller will be more knowledgeable and careful than the typical ebay or CL seller, so start by checking the classifieds right here.
Will you hear a difference by upgrading from the Garrard? Absolutely... 100% guaranteed. Your present table is pretty near the bottom of the pile (sorry, just the truth). The higher up the ladder you go the more improvements you'll hear. In audio there are often diminishing returns as you go upscale, but from the lower rungs anything one does tends to be worthwhile.
FWIW, about 11 years ago I leapt from a late 70s Harmon-Kardon/Rabco rig (which was better than your Garrard) directly to a ~$4,500 table/arm/cartridge - my first high end rig. The improvements were staggering, breathtaking, blah, blah. Even if you don't have that sort of cash to spend at the moment, the principle is worth noting: when upgrading from fairly modest equipment, make the biggest move you can afford. This is a another argument for considering a used (but well loved) rig.
Happy hunting, and welcome to vinyl madness!
P.S. If you actually have records (as opposed to music) from the 1920s, they'd be 78's. Even if the table has a 78rpm speed, those discs require a different cartridge and phono stage than LPs.
