So I salvaged an old turntable/vinyl player from my garage...

Did you replace the stylus? If the technics amp is  a contemporay of the turntable and you've plugged it into the phono inputs then a new stylus / cartridge would be the most obvious thing as it sounds like  yours is completely shot.

You can find the details of the deck and cart including the manual here

http://www.vinylengine.com/library/technics/sl-3.shtml
 

http://www.musonic.co.uk/eps24es-elliptical-stylus-p-15737.html have a choice of the eliptical (ES)  or conical (CS) replacement stylii for your Technics cart. The Eliptical will be better quality but maybe it's worth just getting the cheapest one first to make sure the deck is all working.

The cart is a T4P / P-Mount so there are still a fair number of these around from brands like Stanton, Ortofon, Shure and Audio Technica scaling upto hundreds of pounds but as I say make sure the deck is working properly first before contemplating upgrades.

A good simple tweak is to get some anti vibration feet to go under  the hard original  ones from here http://www.quietpc.com/gb-en-gbp/products/caseaccessories/acoustifeet and don't be tempted to put the deck on top of the amp, rather it needs it's own solid surface away from other kit to work well.

These Technics linear trackers are some of the best decks of that type made and are very cool having all the auto functions which you won't get on new decks today so well worth getting going.

Certainly look at the cartridge and the wiring, all the way from the cartridge to your amp.  I did confirm that your amp has a Phono input, so you are good there.

With vintage gear, make no assumptions, and follow connections from the music to the speakers.  A failed phono stage in your amp could be the cause as well.  Checking that would involve testing your TT on someone elses system, or borrowing an external phono stage.

Nice Table, should be fun.