A decent set of calipers and a 1" micrometer are pretty much requirements, although you can do a lot with just the calipers.
A hammer with brass and/or plastic faces is useful and a good learning project. Always try to push yourself just enough that some small, recoverable, mistakes are inevitable - you won't learn by picking stuff that's too easy nor stuff that's completely outside your skill set. Hammers are great because you can make them fancier, try for close thread fits, and always, always, work on your finish.
Other great projects that can be scaled to your abilities are machinist's jacks, workstops (vise mounted and table mounted), indicator holders for your lathe toolpost, a lathe tool height gauge, organizers / holders for your tailstock or mill spindle tools, a finger plate, wire bending fixtures, soft vise jaws, any number of clamps and hold-downs, replace any cheap knobs or levers on your tools with quality custom knobs, something to adapt an indicator to your tailstock, a tailstock tap and/or die fixture, custom work light mount, a grinding table or other fixtures for grinding HSS toolbits on your bench grinder, a depth gauge - these are all useful additions to your new shop and can be made as simple or as complex as your skills allow.
A caution on tool recommendations. There are a few tools that are pretty fundamental - like calipers and a mic. Most tools, however, have a utility based upon what YOU work on and how YOU solve the problem at hand. I strongly recommend trying to get by with whatever you have and only buying tools when you find yourself *repeatedly* wishing you had something. This advice will necessarily slow you down on some projects, but I can promise you it will save you a lot of dollars and regrets over time. I still fall prey to the odd recommendation - and don't get me wrong, some of the recommendations I've received have been great - but I've also frittered away money on "indispensable" tools that I end up just not using, or rarely using.
Finally, do your research. Check sizes, machine power requirements, and suitability. This forum, and the internet in general, are great places to help you avoid costly mistakes like purchasing tools of inadequate quality or ill-suited to the work you do and/or the tools you own.
GsT