Curt
Capt. Eddie jokes that you can use a brick to sharpen tools if you know what you are doing.
But seriously, while a slow speed grinder would be nice, there are lots of turners (myself included) who use inexpensive high-speed grinders. The main thing to be aware of is that a high speed grinder will remove metal faster than a low speed grinder. All that really means is that you need to use a MUCH lighter touch to avoid heat and unnecessarily grinding away good steel. The one thing I would suggest is that the standard wheels that come with grinders sold in big box stores are pretty coarse, and you should upgrade to a higher grit aluminum oxide wheel as soon as you can. 80 grit is fine.
You can also use oil stones to sharpen your tools (lubricate with a light machine oil - Three-In-One is a widely known brand). That's a slower approach, but slow is good because it also means that you will sharpen more carefully and remove less steel, and your tools will last longer. Alternatively, diamond stones are great for sharpening - use water with a little dishwashing detergent as a lubricant.
You can use also sanders (belt or disk) to sharpen tools. In fact, you can easily (and inexpensively) make a sanding disk that you run on your lathe that will do a fine job of sharpening tools.
Many turners use grinders or sanders as their primary sharpening devices, but then 'touch up' the edges of their tools using inexpensive diamond cards or paddles. That's smart - use the grinder for basic shaping, and then use the diamond hone to keep the edge sharp without wasting metal.
If you go with a grinder, you can make your own grinding jig from wood and parts from the hardware store.
The main thing is to learn to turn. If you have to start out with less than ideal tools - that's fine, you can always upgrade later.
And if those options aren't good enough, there's always Capt Eddie's brick.