Speaking as a relatively newbie, the things i've found most needed are as follows.
Micrometer (digital preferred so you can switch between metric and imperial)
DTI and magnetic stand
Verniers for rough checking (a cheap Lidl set will do)
Dead centre, Half centre, and live centre
HSS tool set, usually around x8 tools, this gets you used to the shape of the tools and their uses,
A bench grinder and oil stone, to grind the set above and hone them, later to go onto freehand grinding your own tools.
Drill set 1-10mm or imperial equivelant to get you started making holes in things, larger holes can be bored to size.
ME taps and dies and correct tapping drills of a size needed for a project, along with holders. I started with Whitworth which was a mistake on my part.
Four Jaw chuck is very helpful!! if a little fustrating to use at first. It does click rather quickly.
Imperial to metric conversion chart, (helps get your head around both sizes)
Slip gauge set to adjust the centre height of your tool, once the pip is gone your there.
As for projects, a tailstock die holder is a very useful tool, which will find use quite often.
To start out, practice turning a piece of bar to a set diameter. The more swarf you make the more you learn until eventually it clicks what you need to do. removing half the required amount of material. This also gets you used to turning and what your machine tells you as it machines. Vary the depth of cut when the lathe sounds like its groaning, back off. Look at the finish your getting, HSS tools vary greatly on how they cut and they perform different. Deeper cuts make for quite a lot of spring on material if its unsupported. So a final pass on the material at the same setting can relieve that spring for a parallel bar.
Importantly as mentioned already, think ahead at what you want to build, list the tools required to build it and aquire just those tools per project, quite a lot of the time very clean fully equipped home workshops see little use, The messy ones are where a lot of things seem to get built.
Lastly remember to stay safe at all times, safety specs on as soon as you enter the workshop, roll those sleeves up, it will become habbit, make a hook for your chuck key, if its not there don't start your lathe!