There will be as many answers as there are members.
So, your aim should be to ultimately get to the point of being able to free hand sharpen your chisels. But, for most people who haven’t the benefit of a tutor, the question is how to get there. High speed bench grinders are one of the most dangerous bits of kit, very easy to use incorrectly and have a stone shatter. But, a very quick way of getting a blade ready for honing. for someone new I would not recommend them.
There are belt systems such as the Sorby or Axminister version, these are within your budget. They produce excellent results very quickly and have a wide versatility of what they can sharpen. Belts are expensive, but last a good time. Most turners have one. Excellent bits of kit, will sell well should you decide to move on.
There are numerous media for hand sharpening, probably using a jig initially to find out what a well sharpened chisel feels like. You can sharpen anything this way……but without a jig, needs practice and also a knowledge of what your trying to achieve. Probably the cheapest solution, and with say an inexpensive eclipse or similar type jig easy to sharpen chisels with.
The Tormak is an excellent system, it’s IMO impossible to injure yourself with it as it turns slowly. The jigs are excellent and can be used with high speed grinders or belts. The instructions and video are excellent sources of knowledge, but you can get this from many other places. They are ‘slow’ for shops trying to earn money off the tools, but for a hobbyist this isn’t an issue.
There are no right or wrong answers. The Tormek will serve you well, it will sell easily should you no longer want it, they are valued. It will get you excellent results to allow you to focus on learning how to use the tools. If woodwork becomes a passion chancers are it will spend more time in storage than use as you progress to other sharpening systems. If you don’t, it will be great to keep your tools sharp without having to learn any techniques.