What are the "must have" tools for milling and turning?

[m_c]

You can spend a fortune on tools, and still not have the one needed for a job, so I wouldn't even contemplate trying to cover all eventualities!

Alot will depend on the size of machines you have, but here's a quick list I can think of-
Mill
Clamp set - the usual generic kit that nearly all suppliers sell, in whatever size that fits the T-slots
Vice - something nearing the capabilities of whatever machine you have, and something smaller if you have the funds. Large stuff you usually clamp straight to the table, but using a vise is normally easier, however mounting small stuff in a large vise can be a pain. If you have a rough idea of what you're likely to machine, use that as a guid to what vise to buy.
End mills - For steel I usually only have 4 sizes. 12, 10, 6, and 4mm. Anything needing another size I just order when needed. I also have a some larger indexable endmills which I do use a fair bit, but that's only any good if you're machine is large/powerful enough (I picked mine up of ebay a while ago for a pretty good price)
Boring head - pretty much essential if using a manual mill and need to bore something.
DTI - pretty invaluable if setting up to an existing hole/truing to a surface.
Edgefinder - as the name implies, pretty essential for setting up from a known edge
Wobbler - handy if you need to pick up marked lines

Lathe
Other than the usual cutting tools, I can't think of much.
Personally I use index tooling for 99% of stuff, with HSS saved for doing profiled stuff, however it can work out expensive.
An indexed parting tool is a revelation for parting off, compared to using HSS.