Welcome!
If you want dead centres, try to find some old Mores Taperr drills, that bhave been discarded because of being, redundant, too short or wrong size.
The important thing is that the Mores Taper is not damaged. (This is your location! )
You hacksaw off the flutes.
Then mount in the Mandrel.
Set the Topslide over to 30 degrees.
With a sharp tool, set to centre height, start taking small cuts. As you take more cuts the taper will begine to take shape.
How do you know when the tool is set on centre height? If, when you face across the end of a bar in the chuck, there is a "pip" in the middle the tool is NOT on centre height, nor will it cut properly.
When you have finally produced a sharp point; you have your first dead centre.
Repeat and you have one for the Headstock and the Tailstock
If the Tailstock barrel is a smaller Morse Taper, (Say 2 MT where the Headstock is 3 MT) you need to find a drill with that taper, and buy a 3 – 2 MT sleeve., so that it can be machined in the Headstock.,
You might, eventually, find a need for a third!
Remembering your budget constraints, I would advise some purchases.
1 ) A set of Zeus Charts.
2 ) Ian Bradley's book, "The Amateur's Workshop" This will provide a lot of information, including how to set up your lathe.
3 ) Use High Speed Steel tooling. Cheaper than carbide tips, and can be sharpened at home. One toolbit will cost about the same as one Carbide tip, but can be used many more times. Grnding tools, will teach you a lot.
So now, I'm advocating buying a Bench Grinder, A 6 " will do, and last for years..
These things are capital investments (Except the HSS toolbits. They will eventually wear out. If you want to support home industries, get a Diamond turning tool from Eccentric Engineering. Costly, but GOOD )
You will need some measuring equipment. A 6" / 150 mm Digital calliper can be bought from a discount store for about £10, so possibly Au$ 15? This will allow you to measure outside and inside diameters and depths, upto 150 mm, and can convert from Metric to Imperial at the touch of a button.
As you gain experience, and confidence, you can expand / improve your equipment, as the budget allows.
Some kit may seem expensive, but it is worth buying good equipment. Buy cheap, buy twice tends to be true.
But you are not looking for Industrial quality and precision- they COST.
You will not be working to micron limits. Neither your equipment, nor your experience, will allow that.
If eventually you can work to make things that fit, within 0.025 mm ().001" ) be satisfied.
You can save money by making some of your own tools. A Centre Height Gauge, a Mandrel Handle, Tailstock Die holders, as instances.
Better to make mistakes on a bit of steel bar than on an expensive casting from a kit!
Taps and Dies will soon figure on your "Wants" list
Buy the ones that you will use most often.
You can spend a lot of money on different types (Whitworth, BSF, BSP, UNC, UNF, BA, Metric, to name some ) so only buy the ones that you are going to use most frequently.
Ditto for drills. If you are going to work in Metric, you may never find a need for Number or Letter drills..
Eventually, you will want a 4 jaw independent chuck, to hold square or irregularly shaped objects, or when you deliberately want to drill /bore something off centre.
This will let you in for Dial Test Indicators (You will find a use for .both Plunger and Finger types. Finger tend to be smaller range, and smaller divisions than Plunger. ) and a Magnetic Base to carry the DTI.
You can use a finger clock internal or mexternal, but a plunger cannot do intrnal, without a fairly rare attachment.
Finger clocks are often 0.0005" per graduation, where Plunger types are often 0.001", (or the Metric equivalents ) but have longer travel.
So horses for courses!
HTH
Howard