Nickademusss,
That's a beautiful Unimat. It looks very well cared for and you are lucky to get it in this condition.
What other metalwork have you done?
Not much, besides automotive body work, drilling cutting...I have done allot of woodworking and my degree is in electrinics
Calipers? Definitely the inexpensive 6-in. dial or digital calipers are good to start, put plan on upgrading. You will need to get a micrometer as well. One-inch or larger or a set will need to be based on the size of metal you want to turn with that Unimat. Everyone here has an opinion: Mine is to get a one-inch (25mm) to start then add more sizes or upgrade after you have worked for a while.
I will add a micrometer to the list 
Do toolbits come with the lathe. Buy some good HSS ones that are already sharpened and some blank ones to sharpen. If you don't have a grinder with a good wheel for sharpening them, then get one of those or buy new wheels for your grinder. Get a wheel dresser. Some like belt sanders or Dremel tools for sharpening: Use them if you have them. You will find that you will need good edges on your toolbits or special toolbits from time to time and it is great to be able to sharpen them or make them as you go. Augment the grinder or belt sander with a bench stone or diamond hand sharpeners.
I do get a set of bits and I do have the equipment to sharpen them.
Get High-quality Drill bits for that tailstock chuck. Also get a center drill or two. When you have a project planned that requires precision-sizing of holes buy a high-quality chucking reamer or two in the size or sizes you need.
I bought a very nice drill index a few years ago, but I agree I need to add some center drill bits
Buy some known-quality material for practice, as well as whatever you have. If you want to make really nice-looking turnings right away, get some 360 brass. 6061 aluminum is also good for first-practice. You might also like to try some plastic just to see what it is like and plastic won't do much to the edges on your toolbits. Acetal (Delrin) is a plastic that is used by people like us.
I have been hoarding bits of aluminum and their is a nice metal shop near me that I can get scraps from.
Start looking for plans for your first project: a Bar-stock project is easier because you can keep making a part until it is what you want and you can also experiment with different materials. I defer to others here to recommend a first project.
I thought about making a wobble type steam engine first, or a simple castings kit from say PMR
You may want to get something to lubricate your cutting tools in action. Some like WD-40, Some like tapping fluid, some like just about any other product out there. I suggest trying small amounts of applied with a brush. I defer to others on their opinions. Please note that brass and plastic can be turned easily without a lubricant.
Definitely get eye protection if you don't have it already. You will need chip brushes (cheap paint brushes) to clean up around your lathe and work area. Most of us use shop vacuums as well. If you're thinking of using compressed air for cleanup you will find that that causes more problems and makes a larger mess. I also have several bench brushes and different sizes of dust pans for bench cleanup.
You will find that you are never finished equipping your shop. My two cents worth is that you try to budget for periodic purchases as you go along and as you learn you will decide what will come next. You will also be able to decide the level of quality you need for what you are making and adjust your purchases accordingly.
I am sure others will chime in here, but I'll just throw out a few more items in a list for your consideration:
-Abrasives for finishing ("Sandpaper")
-Files (and handles)
-Machinist squares and scribers for layout work
-6-in and 12-in precision rulers
-Dial Indicator / Dial Test Indicator
-4-Jaw lathe chuck if you don't have it.
-Taps and best-you-can-afford drills in sizes for those taps. (Plug and bottoming style taps.) You don't have to buy a set if you'll only use three or four sizes.
-Drill rod for precision shafting and using as standards
-Really good hacksaw blades, whether you are using a hand or a power hacksaw or bandsaw, you will need good blades and you'll change them more often than you may have before.
If you try to do milling with your lathe, or covet a mill:
-End mills and holders or collets to mount them
-Clamping set and clamps for securing the work
--ShopShoe
