Best way to fit these parts together.

[Highsider]

A key way will prevent tortional slippage and a Cross pin will resist axial slippage. If you are satisfied with the location of the 3/8" drilled hole, drill it out to 31/64" and then run a well lubricated half inch reamer thru it. Then turn and polish the shaft end to .0002" or so tighter (larger) than what the reamed hole came out. (Requires careful measurements). Key
seat both the hole and the shaft if you want it keyed and deburr them. Mount the key in the shaft and chill it(a freezer will do, but CO2 would be better.) Heat the reamed piece in an oven or with a torch if you can control the heat well enough to keep it from discoloring. Clamp the shaft in a padded vise with the key in the vertical position and using hot gloves, line the keyway up and slide it together. Work quickly so they don't transfer too much heat and stick before they're where you want them. Two ten thousandths of an inch interference fit is about standard for 1/2" diameter interference fits. If you're going to cross pin the fit, do it last. The half inch fit is the strongest you can do with what you've got and using a standard reamer. Good luck.

 

[AGCB97]

Depending on the use and torque transmitted between the two, Locktight 620 is strong and very easy. Or I've seen youtubers silver solder things like this

 

[whitmore]

I have a bar that is ~.55 inches in diameter. I have a hole that is currently drilled out to 3/8 with drill bit.

I want to turn down the bar and or enlarge the hole so that have a tight fit that is accurately centered and will resist radial forces. I assume that I would probably want to use a key to prevent slipping.

If you had a suitable tapered reamer, and could match its taper on the end of the bar,
that will center it and resist radial force. Jeweler's wax or other adhesive would
keep it in. If the fit is good, superglue can hold a ton.