How can you connect a better antenna to a radio like this? I don't
mind taking it apart a bit.

Likely the simplest thing to do is open it up and find the connection
to the existing antenna, then add a connector wired to the same place.

Use the TV antenna.
It won't matter too much if you use twinlead or coax. You can just put a "
F" connector or screw terminals into the radio as above. I prefer to use a
300 ohm balun from the 75 ohm coax for isolation though because of the
possibility of AC leakage to the TV chassis.