from the alternator to the battery there should be a pretty thick cable the size of your finger. that cable is what charges your battery. i've seen it before where someone had cut that wire and just twisted it together and taped it up and or crimped it up. even though you are getting continuity with that cable, it still will not charge your battery. if that dont help, and you get your new battery and alternator from pepboys, autozone, or o'reilly, its possible its bad. i went through 4 batterys with autozone, and 3 alternators with pepboys in a 2 weeks. napa seems to have better replacments parts for toyota/lexus. good luck.
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I say leave your alternator on the car, make sure your battery is fully charged, drive the car to autozone or Napa or pepboys or anywhere different that isn't advanced and test your whole battery starting and charging system. If it all passes then try and find continuity between every wire on the alternator from where it starts to where it ends on the alternator connector. If all the wires are good then just start changing relays around.