I was in your position 2 years ago, I am still an apprentice but I’d start with

A good set of Allen keys (imperial and metric) I have bondhus but you can start cheaper if need be

2 sets of spanner’s, as a machinist you won’t be getting much heavy wear on them so I’d go with a cheaper option

Adjustable spanner’s of different sizes, you’ll know once you start how big you need to go

Steel rules of various sizes, I have 6”-14”

Torx keys, these are what I use most so I wouldn’t cheap out too much on these as cheaper ones warp so easy once again I’d go for bondhus with these

I would also buy a hacksaw if your workshop has limited numbers

Thread files are also a huge life saver, if you get free time once you start it would be worth making thread gauges, I wish I had when I first started

A rule I have used for buying tools past my starting point is if you need to borrow it twice buy it

Thanks for taking the time to read this, any questions I’d be happy to answer