Sharpening a straight razor isn't that big a deal. If you can sharpen a kitchen knife, you can sharpen a straight razor.
In general, if you can get away with stropping dry or with compound, use that till you can't. If you have to hone, the preference now is toward water stones, but a good translucent Arkansas will work well too. The important thing to remember to keep the angle consistant. It's actually easier with a razor than a knife, because they are made to sit flat against the stone, and the spine acts as an auto correcting angle guide.
Another important thing is not to let the edge chip or crack, this is an awful thing to have happen, as you could loose the blade. It usually happens when a razor is set down roughly or falls, but I have seen it happen from using old or dirty hones.
If you're just starting out, something like 5/8 or 6/8 with a regular hollow grind is a good starting point for most people. But if you have a wiry or heavy beard, you might track down a flat ground "wedge" type razor.
Tip style is a personal thing, but I like the square tip better than the round safety tip, as I find it better for detail work. They can be a bit nicky around ears, if you're not careful.