I’m frugal so…jmho
I used a tube type that looked almost identical for 25+ years. Until you want to start larger bowls it should work fine. Not perfect but still fine.
I would pull the drive center and take it to the hardware store. Get a short bolt of soft metal to fit (will still be a lot harder than wood). There should be a setscrew. Cut the bolt head off, insert it, and use a file with the lathe running leaving about ¼” exposed. A new drive center for 15 cents.
A live center for the tailstock would be great. However, dead centers were used for a very long time with no problem. I would try to get by in the short run with the dead center, at worst it will scorch the wood a little which will probably be cut off anyway.
As to tools, an inexpensive set may not be bad. Normally you get a skew or two (1/2 & 1”), a spindle roughing gouge, one or two spindle gouges, parting tool, and a couple of scrapers. I have heard a lot of praise for a $50-60 set from either Harbor Freight or Grizzly, I can’t remember which. Good thick metal and held an edge well, these reviews were by people who turn almost every day.
I’m only a weekend turner and for spindle work use the set of Sorby’s I bought in the 80’s.
One big $ is in the chuck (min $100) and you will want one fairly soon probably even though it is not necessary for most spindle turning. Note: Spindle work does include boxes, pepper mills, tops, bottle stopper .... anything turned endgrain orientation. Nova has some very good prices on their reconditioned with the G3 at $75 or the SN2 at $100.
Another big $ (for bowl turning) is one or two bowl gouges. Some “sets” include a bowl gouge. Hopefully you already have a grinder. You will want a grinding jig for the spindle or bowl gouges (think $100) or there are plans on line for making your own.
They who can give up essential liberty to obtain a little temporary safety, deserve neither liberty nor safety. Benjamin Franklin