So I Went Cheap

If you have a set of regular sized tools, I would use them before jumping into buying miniatues. I've turned small things with regular tools and have never felet I needed a smaller tool. Of course, this depends on just how big your tools are and if you have some smaller ones rather then all big ones. Also, a larger, heavier tool will reduce vibration.
If you find you need a smaller gouge or whatever, you can consider just buying that tool to go with the larger tools. I think LV gives the sizes of the tools. You should also be aware that UK tools use the flute size as the measurement of a gouge  size where in North Americal it is the diameter of the shaft. 
Are your tools carbon-steel or high-speed steel (HSS)? You do not need a slow-speed grinder to sharpen lathe tools but if they are carbon-steel you have to  be careful you don't overheat them (turn blue is the usual symptom). This means a light-touch and some cooling water. HSS is quite immune to overheating so high-speed grinding is not a problem. HSS should not be quenched with with water (can produce cracks). If you do not know what type of steel you have, Google something like "metal spark test" although I suspect being a metal lathe owner this is not likely a problem.
You certainly can get a good edge with a stone and a diamond hone but the nature of spinning wood on a lathe can dull a tool quite quickly so a usual concern is spending too much time sharpening which you would do the the stone/hone. OTOH, some turners do hone some tools after grinder sharpening especially skew chisels and others don't. Another approach is to sharpen the tool on the grinder and then touch it up a few times with the hone before returning to the grinder. Really, all of this will ultimately depend on how you find the process.
The grey carborundum wheels that usually come with grinders are not seen as a good choice but they do work. Sharpening, is nothing more than removing some metal and anything that does that will sharpen the tool - it's just that some things work better/easier than others.
There is a movement to CBN grinder wheels for turning tools and they are good although expensive. They will last for years and never need dressing like a stone wheel. I have found the Norton Blue wheels sold by LV to be good if you don't want to go to CBN. They work well and are hard enough not to quickly groove. LV also sells white wheels but IMO they are very soft and groove too quickly - probably better for carving knives.
There is a lot of buzz about carbide tools because they don't need sharpening (replace cutter instead) and they are easier to use. Yes, they do cut but their cutting action is a scrape rather than a shear cut and the surface is not a good and may have serious tearout on some woods. Most turners recommend to learn how ot use the regular tools but the choice is yours. There are shearing carbide tools such as sold by Hunter but they are not cheap and may be more oriented to hollow-form work. I admit I have some regular carbides and do find them handy for some quick job where I'm not overly concerned about finish. In general, scraping cuts work best on real hard woods (the harder the better) and tend to be worse on softer woods.
You are absolutely right about the cost of the lathe being swamped by the cost of the accessories - but buying new stuff is part of the fun!
The above is my opinion and it is only one in a big sea of turner's opinions, I'm sure you will get others both for and against!
HTH and Good Luck

I agree wkth everything that @Bill Howatt said.
I wrote a longer response and it disappeared !!!  Guess I clicked on the wrong button again.