I'm another new starter, so my experience doesn't quite match up with most of the people on this forum, but I've already learned a few things that I wish I knew a month or two ago.
First, I budgeted £200, then ended up spending double that on a better lathe than I was originally eyeing up. That was probably the best decision I could have made. Get the best you can reasonably afford.
When it comes to sharpening, I researched like crazy and came to the conclusion that there are equal amounts of people making good work sharpening tools on budget grinders as there are using Tormeks, Wolverines and the like. I bought the Screwfix Titan 8" bench grinder, fitted it with a white wheel from Abtec (Although before that arrived I was getting good enough results if I was careful with the wheel that came on it) and I built Keith Rowley's homemade grinding jig very badly with some scrap and about 70p worth of bolts. It's not pretty but it does the job no problem. The Titan grinder is ten times better than most tools at the same price point, and I struggled to find an eight inch grinder with a motor that powerful for that cheap. It does seem to be hit and miss whether you get one with wheel wobble or not but luckily mine was minor and dressing the wheels properly with a diamond tool made it barely noticeable, if at all.
My last good decision was a honeywell bionic face shield and a 3m mask with changeable P3 filters. I turn in a fairly well ventilated environment so I don't feel the need for a powered respirator. I've yet to blow one piece of sawdust out of my nose after turning.
If you think you'll need a chuck at all, get the best you can find. I thought I could use the budget piece of junk that was thrown in free with my lathe and I still haven't found all the pieces of one bowl that shattered and ping-ponged around the shed. I'd like to sell the chuck, but I don't think I could bring myself to subject someone else to something that dangerous. I'm saving for a Nova or a Patriot.
Finally, I'm not sure about anyone else but if I had a decent amount of money to spend on one tool I'd pick the spindle gouge. I have a mid-range starter set and it's all good, even the spindle, but I do 80% of my turning with it so I'm on the lookout for a better one. I thought the bowl gouge would be most important, but honestly I've finished bowls using just the spindle gouge and round ended scraper. My Robert Sorby bowl gouge gets more action rounding down blanks during faceplate work.