https://smile.amazon.com/Barfly-M37101-Cocktail-4-Piece-Stainless/dp/B07BB9VB5P

This is pretty much exactly what I would recommend, and what I got. The barfly shaker is definitely what I recommend for its price and its quality.

Pretty much any jigger will do, but obviously imperial/metric depends on your locale, and straight/bell shaped is personal preference. I also definitely prefer a 1/2oz jigger instead of a 1/1.5oz model. Also, check what lines are on the inside. Mine has (counting the full bells) 2, 1.5, 1, .75, .5, and .25. I can definitely see not having any of those to be irritating at the very least.

Barspoons are entirely preference. Honestly, I'd just get whatever. Just measure the spoon bit so you know :)

Hawthorne strainers just need to fit your shaker tins and mixing glass (if you buy one, I wouldn't yet). Look for one with a nice, tight coil, like that one in that set. The crappy ones will look very spindly, and if you can't get a decent one for whatever reason, you can wind two springs together to make it work with the crap ones.

A mixing glass can be very nice to have, and it definitely looks better if you're entertaining or working. Functionally though...it's a glass. Use one of your tins or something.

I was just about to recommend looking at Barfly as a slightly cheaper Cocktail Kingdom alternative, only to find out that they offer the basic set of tools mentioned above for $53.50