403's are nice machines! I love the slant shank Singers. I have one of these, too.
Needles - good news! Standard sewing needles fit this machine.
It takes class 66 metal bobbins but beware - DON'T buy new metal bobbins at a place like Joann's - even Singer brand. New class 66 bobbins are pretty awful and many won't even fit or work in your machine (discovered that for myself the hard way!) Buy vintage ones from eBay if you can find them.
If you need new feet for the machine, make sure you're getting "slant shank" feet. They're somewhat unique but not really difficult to find.
The cams your machine uses are also very common! They're called "top hat" cams and the same cams that fit in your 403 also fit in most 500 and 600 series Singer machines, so they're pretty easy to find. Watch a few listings on eBay and you'll get a feel for what's a bargain. Your machine also has several built-in stitches, so depending on what you do you may not really "need" the cams anytime soon.
Sewing machine oil is great for cleaning the outside of the machine; baby wipes are also fine.
Definitely read over the manual and make sure you know where all the oil and lubrication goes! These old machines are tough but they still appreciate a little oil and grease in the right spots. Make sure you use oil designed for sewing machines, no 3-in-1 oil or WD40. The manual should walk you through what goes where, but post here if you get stuck.
If you need bits and pieces (including probably your missing cord), try Sew Classic. They carry a lot of the little bits and bobs you'll need for your machine.
Congrats! You have a very nice machine and you got it for a steal!